Estimating Economic Relationships under Measurement Error: An Application to the Productivity of US Manufacturing

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1 Estimating Economic Relationships under Measurement Error: An Application to the Productivity of US Manufacturing T. H. Y.Tran, A. N. Rambaldi,A.Peyrache School of Economics, The University of Queensland, St Lucia, QLD 407. Australia This version: 1 February 015 Abstract We propose an approach to the problem of measurement errors that evokes long established but rarely used results about the identifiability of the error-in-variables (EIV) models. Our approach uses the dynamic structure of the true series and measurement errors to identify the parameters of interest. The dynamics of the underlying time series are introduced into the model using a structural time series approach and the identification of the parameters of interest is achieved by a simple property of the multivariate normal distribution. This modeling framework has several advantages. The first is the possibility of incorporating more flexible components of the time series being studied, such as trends, cycles, and seasonality. The second is that the model allows for a non-zero correlation between the measurement errors of the variables involved in the structural relationship. The third is that using a multivariate normal distribution to derive the structural relationship between variables allows for the time-variation in the relationship, i.e., in both slope and intercept parameters. We prove two results to show our estimator can identify the structural parameters, provide a simulation exercise and an empirical illustration using data from the Bureau of Labor Statistics to compare our findings to those recently presented in Diewert and Fox (008), where they raise the issue of severe measurement error and the endogeneity of inputs and outputs. Keywords: unobserved components, time-varying parameters, least squares bias JEL Code: C18, C3, C51 1 Introduction The paper proposes an approach to identifying and estimating the parameters of a structural relationship between variables measured with errors. As is well documented, measurement errors are prevalent in many settings. For example, survey data of earnings and work hours are highly susceptible to measurement errors (Devereux, 001). Falk and Lee (1990) suggest that measurement errors can explain rejections of the permanent income hypothesis. Van Beveren (01), Klette and Griliches (1996) and Ornaghi (006) show that the use of industry-level prices to The authors would like to thank Erwin Diewert, Andrew Harvey, Adrian Pagan, Robin Sickles, attendees of the CEPA 1-Day Workshop (3 October 014, held at UQ) and the Advances in Empirical Finance, Financial Econometrics and Macroeconometrics Workshop (18-19 November 014, held at UQ)for many constructive suggestions in our earlier work on this project. All errors remain ours Corresponding author. a.rambaldi@uq.edu.au 1

2 deflate firm-level sales in order to obtain a proxy for firm s output quantity instead of the real quantity likely results in biased input coefficients and unreasonably low estimates of returns to scale. Diewert and Fox (008) (D&F hereafter) derive theoretically consistent aggregation formulae to construct Törnquist indexes of input and output growth, as well as a relationship that links them so that technological change and returns to scale are parameters of this relationship. D&F make strong arguments for the appropriate treatment of measurement errors in the statistical analysis of the relationship. Their arguments are that some of the assumptions necessary for the exact relationship to hold, such as the constant markup factors, the cost-minimizing and profit maximizing behavior of firms, the functional form of the cost function etc., do not hold empirically. They also argue that methods of aggregation used by statistical agencies might not reflect the specific methods of aggregation that the exact theoretical results rely on, and therefore both the input and output growth measures used to estimate the model are likely to suffer from measurement error. Although acknowledging these potential problems, D&F did not attempt to address them empirically, and produced their estimates by least squares arguing that this provides reproducibility of the results. In the empirical section of this paper we use D&F s model and data to illustrate our proposed approach and explore whether the estimates differ substantially when we allow correlated measurement errors in the input and output index series. In addition, our estimator makes no exogeneity assumptions about the input and output series, which are treated symmetrically. D&F discuss the issue of which is the right regression (see pp ) in their paper. They settle into using output growth as the dependent variable in their regressions. While it is a well justified argument to regress output growth on input growth, our estimation approach that does not place strict exogeneity assumptions on the input growth series providing a more appealing estimation alternative as well. Our proposed approach to the problem of measurement errors evokes the long established but rarely used result about the identifiability of the error-in-variables (EIV) models. That is, in the static models with errors in variables, the parameters are unidentified unless there is available prior information about the variances of the measurement errors, the reliability ratio (the ratio between the variance of the measurement error and that of the true data), or additional measures which can be used as instrumental variables (see Fuller (1987); Reisersol (1950); Kapteyn and Wansbeek (1983); Buonaccorsi (010)). However, identification is also possible by the use of some dynamic structure, either in the true series or in the measurement errors, as firstly suggested by Maravall and Aigner (1977). Identifiability conditions of such models are studied in Anderson and Deistler (1984), Solo (1986), Nowak (1993). More recently Komumjer and Ng (014) study identification criteria in a model with serially correlated measurement errors. The present work uses these existing results about the identifiability in an attempt to model the relationship between variables which are measured with correlated errors. The essence of our modeling approach is to incorporate the dynamics of the underlying time series using a structural time series framework (a comprehensive treatment of structural time series models can be found in Harvey (1989) or Durbin and Koopman (01)). Furthermore, as suggested by Harvey (013), using a simple property of the multivariate normal distribution, the structural parameters in a regression relationship can be identified through the mean vector and the parameters in the covariance matrix. This modeling framework has several advantages. The first is the possibility of incorporating more flexible components of the time series being studied, such as trends, cycles, and seasonality. This differs from the dynamic structure commonly specified in the literature, i.e., ARMA, VARMAX, etc. The second is that the model allows for the non-zero correlation between the measurement errors of the variables involved in the structural relationship. As it will be shown, this correlation contributes to the bias of the OLS estimator and therefore, ignoring this correlation might have significant consequences on the estimates of the parameters of interest. The third is that using a multivariate normal distribution to derive the structural relationship between variables allows for the time-variation in the relationship, i.e., in both slope and intercept parameters. In this paper we provide a simulation exercise and an empirical illustration for the case when the intercept in the structural regression relationship is time-varying and the slope is constant. We also put forward a proposed estimation strategy for the case when both the intercept and the slope are time-varying. This is work in

3 progress. A brief review of identification strategies in the presence of measurement errors The literature on the treatment of measurement errors is vast and it is not possible to provide a comprehensive coverage. This section is only intended to cover some of the most prominent works that are of relevance to the approach proposed in this work..1 Static EIV Models This review is largely based on Fuller (1987) and we use his notation. We commence with a generic specification of the relationship between the variables of interest. Consider the variables X t and Y t, Y t = x t + e t,x t = x t + u t (1) (x t,e t,u t ) 0 N [(µ x, 0, 0) 0,diag( xx, ee, uu)] This model assumes x t is measured with errors u t and the observed series of this variable is X t. It is assumed that e t, which can either be measurement errors in Y t or the error in the equation, is uncorrelated with u t. The model is clearly unidentified because there are six parameters, ( 0, 1,µ x, xx, ee, uu); however,sincey t and X t are normally distributed, their distribution is completely characterized by the first and second moments, a total of five parameters. More general results of identification are established by Reisersol (1950) and Kapteyn and Wansbeek (1983). Reisersol (1950) proves that in the bivariate model in which the measurement errors are jointly normally distributed, a necessary and sufficient condition for to be identifiable is that the joint distribution of X t and Y t is not normal. Kapteyn and Wansbeek (1983) generalize this to a similar result in the case of higher dimension. With the specification (1) identification only becomes possible when some prior information is available. The first strategy is assuming the measurement error in x t is known, i.e., uu can be estimated. The second strategy is assuming the reliability, i.e., = 1 XX xx, can be estimated where XX is the variance of the observed X t. This is also the attenuation coefficient by which the OLS estimate of is biased. Therefore, if this ratio can be computed, we can use it to correct for the bias of the OLS estimator. The equivalent maximum likelihood estimators for these cases are described in detail by Kendall and Stuart (1977). In practice, neither uu nor are readily available, and they are often estimated through replication (Buonaccorsi, 010). One assumes that for each t there are m t replicate values W t1,...,w tmt of the error-prone measure of x. Then we have W tj = x t + u tj ; j =1,...,m t where u tj is the measurement error of the proxy W tj fortheobservationt. The mean of measurement error u t = E(u tj )=0and its variance var(u t )=var(u tj )/m t = var(w tj x t )/m t where x t is approximated by x t = E(W tj x t ) 1 P mt m t j=1 W tj. Asecondstrategyistheinstrumentalvariableapproach. Assumetherearetwoproxies,Y t1 and Y t,available for the variable y t. Then, the model can be specified as: Y t1 = x t + e t1 () Y t = x t + e t X t = x t + u t Assume that xx > 0, 11 6= 0, 1 6= 0and (x t,e t1,e t,u t ) 0 N [(µ x, 0, 0, 0) 0,diag( xx, ee11, ee, uu)]. Under these assumptions, especially the assumption about the diagonal covariance matrix of the unobserved series and measurement errors, we have 3

4 6 4 Y t1 Y t B6 5 4 µ Y 1 µ Y X t µ x ; 4 11 xx + ee xx 11 xx 11 1 xx 1 xx + ee 1 xx 11 xx 1 xx xx + uu 31 7C 5A Given a sample of t observations, the matrix of sample moments about the mean is: 3 m YY11 m YY1 m YX m ZZ = 4m YY1 m YY m YX1 5 m XY 11 m XY 1 m XX11 The model contains nine unknown parameters: ( 01, 11, 0, 1, ee11, ee, uu, xx,µ x ) which can be uniquely identified from nine sample moments. It is easy to see that this method is using Y t as an instrument for the relationship between Y t1 and x t and using Y t1 as an instrument for the relationship between Y t and x t. A different approach was proposed in Marschak and Andrews (1944), Anderson (1976), and Klette and Griliches (1996). These papers take into account the simultaneity and measurement error problem by abandoning the one equation model as in (1) and specifying a more complete simultaneous equation model which identifies truly exogenous variables (measured without error) from endogenous variables and variables measured with error.. Dynamic EIV models Maravall and Aigner (1977) point out that a static model that is unidentified in the presence of serially uncorrelated measurement errors could be identifiable when the model has a dynamic structure. Intuitively, if the measurement errors are white noise but the model has dynamics, the model can be identified by the suitable choice of instruments such as lags of Y t and X t. Anderson and Deistler (1984) assume that the input and the measurement errors are autoregressive moving average processes (ARMA). Nowak (1993) extends this to a multivariate setting where the underlying series follows a VARMA process. Solo (1986) and Komumjer and Ng (014) consider an ARMA model with exogenous variables (ARMAX) where measurement errors are serially autocorrelated. The closest approach in the group of dynamic EIV models to what we propose in this paper might be that of Sargent (1989), which adopts a state-space system that explicitly allows for measurement noise. He shows how the Kalman filter can be used when the variables in a dynamic model are subject to autoregressive measurement errors. Assume that the economic model takes a state-space form x t+1 = Ax t + t Z t = Cx t The disturbance t in the transition equation is such that E( t 0 t)=q for s = t or 0 for s 6= t and E( t )=0for all t. There is a vector of measurement errors v t that obeys v t+1 = Dv t + t where t is a normally distributed white noise. A data collecting agency is assumed to observe a noise-corrupted version of Z t,namelyz t,governedby: z t = Cx t + v t Using the Wold representation, Sargent transforms the system above with first-order serially correlated measurement errors to one with white-noise measurement errors. Then, the standard Kalman filter applies. In the same spirit of introducing some dynamics into a static EIV model to make identification possible, Harvey (013) remarks that the identification problem can be resolved if the dynamic properties of the components of 4

5 interest are different from those of the errors. An example of this identification strategy is in Harvey (011), where the Phillips curve relationship is built from the joint cyclical components of the unemployment rate and GDP, after trend, seasonal, and irregular components are extracted. This extraction is possible because their dynamics differ from those of the cycles. This idea is similar to that of Maravall and Aigner and the estimation approach also makes use of the state-space representation and Kalman filter with likelihood maximization as Sargent s approach. However, Harvey s approach uses a structural time series framework, which looks at time series data as consisting of different components, instead of a Box-Jenkin type approach, which models data as (V)ARIMA processes. In this paper, we follow Harvey s approach and show that it provides great flexibility in the modeling as well as being relatively easy to implement using existing commercially available packages. 3 Harvey s approach to the estimation of regressions with time-varying parameters Although our first target is not to model the time-varying parameters, the approach suggested by Harvey (013) for this problem provides a venue to incorporate some dynamic structure into the regression model which can be used for identification when measurement errors are present. Therefore, we summarize his approach here. Harvey suggests to model the regression relationship from a multivariate distribution point of view. The parameters in the regression relationship depend on the parameters of the distribution. Suppose that y t and x t are jointly normally distributed with constant means, constant variances, and a time-varying correlation: y t x t N µ y µ x ; y t y x t y x x The distribution of y t conditional on x t is also normal and the regression equation can be written as (3) y t x t = t + t x t + t (4) where t = t y x ; t = µ y t µ x (5) As clearly seen, the structural parameters in the relationship (4) can be identified through the parameters of the normal distribution as in (5). Harvey (013) suggests that we can model the time-varying t through the time-varying t by means of a Dynamic Conditional Score (DCS) process. This is an autoregressive process driven by the conditional scaled score, which is the first derivative of the log-likelihood function scaled by the information matrix. Thiele and Harvey (013) adopt this approach to examine the integration of the Shanghai and New York stock prices over time. The returns of the two stocks are assumed to follow a bivariate t-distribution. Then, the relationship between the two stock prices is modeled through the time-varying correlation and variances in their joint distribution. The DCS approach will be explained in Section 7 when we consider the estimation of the model when both t and t are time-varying. In earlier sections we work with the case when t =. 4 Unobserved components model and estimation In this section we extend Harvey s approach summarized in Section 3 to tackle the problem of identifying the structural parameters in the presence of measurement error. We extend (3) by interpreting (µ y,µ x ) as the unobserved true series and assuming that they follow stochastic processes. Measurement errors are incorporated as 5

6 identically independently normally distributed noise. A general specification might incorporate trends, cycles and seasonality, which might help to better explain the data. We provide this generalization in Section 4.3. However, in order to develop the model and the identification of its parameters, we start with a simple specification where the trends in the underlying series follow random walk processes. 4.1 The model Our target is to model the exact relationship between the underlying variables which are measured with errors and to allow the relationship between them to change overtime. y t = µ yt + v yt (6) x t = µ xt + v xt (7) µ yt = t + µ xt (8) Equation (8) describes the exact relationship between variables µ yt and µ xt. We only observe y t and x t which 0 are contaminated with measurement errors given by v t = v yt v xt, which is assumed to follow a multivariate normal distribution with zero mean and the constant covariance matrix = vy vy vx vx vy vx We assume that t and µ xt follow random walk process, i.e., t = t 1 + t (9) µ xt = µ xt 1 + xt (10) where t and xt are jointly normally distributed with zero means and diagonal covariance matrix = It follows that µ yt = t 1 + µ xt 1 + t + xt = µ yt 1 + yt 0 0 x where yt = t + xt. Thus, y = + x and cov( yt, xt )= x. We can write the structural model (6)-(10) in an unobserved component (UC) representation as follows. y t x t = µ yt µ xt The measurement equations, (11), decompose the vector of observed variables into their underlying trends and the measurement errors. The transition equations (1) specify the dynamics of the underlying trends as random walk processes: The signal vector t = yt µ yt µ xt = µ yt 1 µ xt 1 + yt xt + yt xt xt 0 follows a multivariate normal distribution with zero mean and covariance matrix. y y x = y x x. (11) (1) 6

7 We make the assumption that there is no serial correlation in the measurement error series and the signal series, and these two series are uncorrelated with each other. By (11) and (1) we define the UC representation of the structural model, (6)-(10). In the next subsection, the first proposition will identify the structural parameters in (6) in terms of the parameters in the UC model. The second proposition will demonstrate how measurement errors and their correlation contribute to the bias in the OLS estimator, which in turn helps clarify how the estimation of the UC form can avoid the measurement error bias. 4. Identification of the structural parameters Proposition 1. Assume the data follows the model defined by (6)-(10), and, thus, the equivalent UC model is defined by (11)-(1); then, the structural parameters are identified as: = y x (13) t = µ yt 1 µ xt 1 (14) Proof. Since y t and x t are measured with errors, the structural relationship between them should be derived from their underlying means µ yt and µ xt.from(1)andthenormalityassumptionof( yt, xt ) 0, we can write the joint distribution of the underlying means of y t and x t given past information = t 1 = {(y i,x i ),i=1,...t 1} as: µ yt µ xt = t 1 N µ yt 1 µ xt 1 ; y y x y x x Since a multivariate normal distribution of a vector of variables implies normal distribution of each variable conditional on the other variables, (see Greene (01) for instance), µ yt is normally distributed conditional on µ xt. So we can write y y µ yt µ xt, = t 1 = µ yt 1 µ xt 1 + µ xt + t (16) where t N(0, ) and = y x.matchingthisrelationshipbetweentheconditionalmeanofµ yt given µ xt and the exact relationship equation (8), we can identify the structural parameters as: x x (15) = y x t = µ yt 1 µ xt 1 Remark. The estimation of the structural parameters reduces to the estimation of the covariance parameters and the state vector of the UC model. can be made time-varying by allowing to be time-varying. This case will be considered in Section 7. 7

8 The identification of t and is possible even though no prior information about the variance of the measurement error, the reliability ratio or other extra information is used. Identification is achieved because the dynamics of the true underlying processes are different from those of the measurement errors. This would not be possible in the case where (µ yt,µ xt ) are time-invariant. Consider (17) y t x t µ yt µ xt = = µ yt µ xt µ y µ x + v yt v xt + yt xt (17) where the measurement errors and the signals are normally distributed with the covariance matrices v and, respectively. This is the classical case of measurement errors. Reisersol (1950), among others, established that the normality of the underlying series in a static model where measurement errors are also normally distributed, is what spoils identification unless prior information is available. The identifiability of the UC framework can also be seen by evoking the stochastic properties of the vector 4Y t =(4y t, 4x t ) studied in Harvey (1989, p. 431). We have for the UC representation that 4Y t = t + v t v t 1 is a stationary process. The autocovariance matrices are (0) = + v (18) (1) = v (19) ( ) = 0, (0) It is clear that two covariance matrices of the UC model are identified by the first two autocovariances of the observed variable vector. Another important point to note is that unlike the traditional treatment of measurement errors, the UC approach allows for the measurement errors to exist in both µ yt and µ xt and for them to be correlated. The estimation of the UC model is made easy by writing it in a state space representation and applying the standard Kalman filter coupled with maximum likelihood estimation. We will show the estimation procedure in the next section. However, it will prove useful to consider what the OLS estimator will yield if we apply least squares to the differenced series. Proposition. Assume y t and x t follow the UC model in (11)-(1); the OLS regression of the first differences of y t on x t will return = y x + y x x + vx (1) Proof. From (11) - (1), we write the first differences of the observed variables vector as a sum of the signal and the measurement error vector: 4y t 4x t = yt xt + v yt v xt v yt 1 v xt 1 8

9 Our assumptions of the UC model allow us to write the distribution of the first differenced series as 4y t 0 y + y y x + y x N ; 4x t 0 y x + y x x + x The regression that corresponds to this bivariate normal distribution is given by 4y t = y x + y x x + x 4x t + t () By construction, 4x t is uncorrelated to t (see, for instance, Greene (01)). OLS applied to the first differences series will provide the estimate of : = y x + y x x + vx Remark. Granger (1986) and Phillips and Durlauf (1986) established that if µ yt and µ xt are cointegrated but are observed with error, then the two observed series, y t and x t, will also be cointegrated if all measurement errors are I(0), and there is no asymptotic measurement error bias when the regressors form an integrated process (for a more detailed review of this literature, see Hassler and Kuzin (009)). In our case, µ yt and µ xt are not cointegrated as t is a random walk process. Therefore, OLS applied to the observed levels, y t and x t,islikelytobespurious.for this reason, we apply OLS to the first differenced series, which are stationary processes. The OLS estimator is obviously inconsistent due to the measurement errors variances and covariances, vy, vx and v vy vx. However, there are two special cases where OLS applied to the first differenced series can estimate the true structural parameter. 1. The first case is when there is no measurement error in x t, vx =0. Then the measurement error bias caused by y x and x becomes zero. The expression of in (1) becomes: = y x + v vy vx x + vx = y x = (3) This shows that, in the absence of measurement errors in x t,olsappliedtofirstdifferencescancorrectly identify the time-invariant measurement error is present in x t.. However, this also implies the OLS estimator should be excluded as soon as. The second case is when the UC model is homogenous, p = v.technically,thismeansthatthecovariance matrix of the measurement errors is proportional to the covariance matrix of the signals. Intuitively, this means all linear combinations of y t and x t have the same stochastic properties (Harvey, 1989). When this is the case, the OLS estimator in (1) collapses to the estimator in (13). = y x + y x x + vx = (1 + p) y x (1 + p) x = y x +p y x x +p x = y x (4) = (5) This coincidence is not just a mathematical reduction of the formula resulting from the restriction = p. In fact, statistically, this homogeneity directly implies that x t is exogenous in the sense that conditioning on x t does not improve the information about the structural parameters we are interested in (Harvey, 1989). 9

10 However, the homogeneity assumption is clearly very restrictive. Therefore, in the general case, OLS estimator is inappropriate. Finally, comparing (1) with (13), it can be seen that by selecting relevant components (the variances and the correlation of the signals, not those of the measurement errors), we can extract the bias due to measurement errors and identify the true structural parameters of interest. This subsection has provided the identification of the structural parameters in the exact relationship where the true variables are unobservable and assumed to follow random walk processes. The following subsection exemplify an extension where cyclicality is incorporated into the underlying true series. Seasonality might be incorporated in asimilarfashion. 4.3 Alternative specification for the UC model - trend plus cycle model. An alternative and more general specification from the simple random walk case above (known as a local level model), is to assume the following specification which allows for the presence of cycles (seasonality could also be incorporated): E yt xt y t x t y t x t µ yt µ xt = = = y t x t µ yt µ xt + µ yt 1 µ xt 1 + h i c yt c yt y cos cy c yt = 1 0 ; = c yt c yt y sin cy h i c xt c xt x cos cx c xt = 1 0 ; = c xt c xt x sin cx h i i yt yt xt = E h yt xt = c ; E xt v yt v xt + c yt c xt yt xt (6) (7) (8) y sin cy c y 1 yt + (9) y cos cy c yt 1 yt x sin cx c xt 1 xt + (30) x cos cx c xt 1 xt i h yt xt = 0; (31) Equation (6) decomposes the observed vector into the measurement error vector and the true underlying vector. Equation (7) defines the true underlying vector as comprising of cycles and trends. Equation (8) assumes that the trend vector follows a random walk process with the innovation ( yt, xt ) 0, which follows a normal distribution with mean zero and covariance matrix.equations(9)-(30)specifythedynamicsofthecycles.sincethecycle in each series is driven by two disturbances, there are two sets of disturbances and these are assumed to have the same covariance matrix, as specified in (31) (following the convention described in Harvey (1989) and Koopman et al. (1995)). y and x are the damping factors while cy and cx are the frequency of the cycles in y t and x t, respectively. Given (9)-(31), we can rewrite the dynamics of the cycle vector as: yt xt 10

11 6 4 c yt c xt c yt c yt c xt c xt 3 = 7 5 = c yt c yt c xt c xt y cos cy y sin cy 0 0 y sin cy y cos cy x cos cx x sin 7 6 cx x sin cx x cos cx c yt 1 c yt 1 c xt 1 c xt yt yt xt xt (3) (33) where the cycles disturbance vector follows normal distribution with covariance matrix 3 y 0 y x 0 = 0 y 0 y x 6 4 y x 0 x y x 0 x Although the specification above seems complicated, we can still write the distribution of the underlying vector as a normal distribution with mean vector and covariance matrix as: E y t x t = t 1 = µ yt 1 µ xt y cos cy y sin cy 0 0 y sin cy y cos cy x cos cx x sin 7 6 cx 5 4 c yt 1 c yt 1 c xt = µ yt 1 µ xt x sin cx x cos cx 3 y cos cy y sin cy x cos cx x sin 6 cx 4 c yt 1 c yt 1 c xt c xt 1 = = µ yt 1 µ xt 1 µ yt 1 µ xt 1 y cos cy c yt 1 + y sin cy c yt 1 + x cos cx c xt 1 + x sin cx c xt 1 + c yt 1 c xt 1 c xt 1 11

12 Var y t x t = t yt = + Var yt C 6 xt 5A xt y 0 y x = + 0 y 0 y x y x 0 x y x 0 x 0 0 y y x = + y x x = + c Therefore, applying the result from Proposition 1, we can identify the structural parameters as: = ( y x + y x ) x + x (34) t = (µ yt 1 + c yt 1 ) (µ xt 1 + c xt 1 ) (35) Which model, local level or trend plus cycle, to be estimated to obtain the structural parameter is an empirical question. Therefore, model selection must precede estimation. As the model has a standard structural time series form, i.e., the time series is decomposed into trend, cycles and irregular components, model selection and estimation are greatly facilitated by many existing commercial softwares. In the empirical section of this paper we illustrate using Structural Time Series Analysers, Modeler and Predictor (STAMP). 5 Estimation This section briefly shows how to estimate the UC model to obtain the covariance parameters and underlying mean series required to compute the estimates of the structural parameters (i.e. equations (34) and (35)). When is time-invariant the UC model is readily in a standard state-space form, where the covariance parameters are hyper-parameters of the representation and the underlying means are the state vector, and thus the Kalman filter applies immediately. In the following, we only present the state-space form and the Kalman filter algorithm for the case of the local level model. However, the idea is the same for trend plus cycle model and many other more complicated structural time series models; that is, representing the UC model in a state-space representation so that standard Kalman filter coupled with Maximum Likelihood estimation can be applied to estimate the state vector and hyperparameters (for details and properties of the classical approach to the estimation of a state-space model see Harvey (1989)). The local level form of the UC model is given by 1

13 y t x t µ yt µ xt vy v vy vx where v = and = v vy vx vx = = µ yt µ xt µ yt 1 µ xt 1 + v yt v xt + yt xt y y x y x x The standard Kalman filter works as it is easy to see that Y t = y t x t., µ t = µ yt is the identity matrix), and thus the UC model can be rewritten in the standard state space form: µ xt (36), Z t = I, T t = I, R t = I (I Y t = Z t µ t + v t ; (37) µ t = T t µ t 1 + R t t (38) The Kalman Filter algorithm is implemented by running through the prediction and the updating steps given estimates of v and.letm t t 1 be the optimal estimator of µ t given information up to and including the period t 1 and P t t 1 be the covariance matrix of the prediction error. Also let m t be the optimal estimator of µ t once the t th observation is realized and P t be the covariance matrix of this estimator. The predicting equations are given by: m t t 1 = T t m t 1 (39) P t t 1 = T t P t 1 T 0 t + R t Q t R 0 t (40) Once a new observation is available, the updating will proceed through the following equations: m t = m t t 1 + P t t 1 Z 0 tf 1 t. t (41) P t = P t t 1 P t t 1 Z 0 tf 1 t Z t P t t 1 (4) where F t = Z t P t t 1 Z 0 t + H t. t = y t Z t m t t 1 is called innovation, and used as the driver of the updating process. Starting from the assumed distribution of the initial state vector with mean, u 0,andcovariancematrix,P 0, the KF delivers the optimal estimate of the state vector. It can be shown that under the normality assumption of v t and t, y t is also normally distributed conditional on past information with mean E(y t )=Z t m t t 1 and variance F t = P (y t )=Z t P t t 1 Zt 0 + H t.accordingly,thelog-likelihoodfunctioncanbeeasilyconstructedas L( y t,m t ) = log TY p(y t Y t 1 ) t=1 = NT log 1 T t=1log F t = NT log 1 TX log F t t=1 1 T t=1(y t Z t m t t 1 )Ft 1 (y t Z t m t t 1 ) (43) 1 TX 0 tft 1 t (44) t=1 13

14 This form of the log-likelihood function provides a convenient avenue to estimate the hyperparameters, = { vy, vx, v, y, x, }. Given these MLE estimates, the Kalman filter and smoothing algorithms are used to provide estimates of the state vector (µ yt,µ xt ) 0. The structural parameters in (8) are computed as in (13) and (14). In the next section, we will present some simulations (coded by the authors using MATLAB) to demonstrate the results of Proposition 1 and. We will use STAMP to implement the empirical illustration with actual data (that used by Diewert and Fox (008)) to assess whether there is any evidence of measurement errors in the data and whether ignoring this possibility has affected the results they obtained. 6 Simulations and empirical example 6.1 Simulations We present three simulations. The first considers the case when there is no measurement error in x t,andthetrue data are generated from the UC model. We therefore expect the OLS slope estimate is the same as that estimated by the UC framework (by Proposition ). The second simulation looks at the case when there is measurement error in x t, and by Proposition 1, the estimation using the UC framework is able to estimate the correct parameters of interest (one of which, the intercept, is time-varying). By Proposition we know the size of the bias in the OLS estimate and the simulations confirm this result. In the third case, we generate data from the structural regression relationship, instead of from the UC model. In this case the true DGP is not necessarily multivariate normal. However, we show that the estimation of the UC representation provides the correct parameters of interest. Each simulation consists of 00 replications. Each data set has 100 time periods (after the first 50 observations are truncated to ensure the time-series are not affected by the initializations). Simulation 1: Nomeasurementerrors-TrueDataGeneratingProcess(DGP)istheUCmodel Data are generated to follow the UC model in (11)-(1) with the following hyperparameters v = 0 ; = ; µ y0 µ x0 = 0 5 In this specification, x t is measured correctly. The true values of the structural parameter in the regression y t = t + x t + t are given by = cov / x =4.5/3 =1.5; t = µ yt 1 µ xt 1 = µ yt 1 1.5µ xt 1 Results of Simulation 1 Figure 1 presents the distributions of the OLS and the estimates of computed using expressions in (13) and (14). The UC model is estimated using the standard Kalman filter (SF) as detailed in Section 5, and thus the estimates are labeled SF in the graphs. By Proposition we expect that in the absence of measurement error in x t,boththeolsestimatoroftheregressioninfirstdifferencesandthesfestimatorcanestimatecorrectlythe slope. Both estimators have approximately the same distribution, which is symmetric and centered around the true value, =1.5. The SF can estimate correctly the slope because it disentangles the true structural parameters from the bias due to measurement errors. In this case, no measurement error is present in x t. The SF estimator, of course, remains valid. In contrast, it is due to the zero measurement error in x t that the OLS estimator yields the true structural slope. 14

15 We construct confidence intervals for both OLS and SF estimates in each replication. The rate at which the confidence intervals of the SF estimates include the value true =1.5 is 189/00 = 94.5%. The rate of coverage for the OLS estimates is 186/00=93%. This means we can be confident at the 10% level of significance that both the SF estimator and the OLS estimator can estimate correctly the slope in the structural relationship. Figure 1: Distributions of ˆOLS and ˆSF Figure : ˆ t,sf from the 19th sample OLS in first differences is not able to estimate the time-varying parameter. In contrast, KF and MLE applied to the UC model is able to estimate the time-varying intercept. We construct pointwise confidence intervals for the estimation errors of these estimates of the intercept t. The coverage rate of the confidence intervals over the true values at all points on the series ranges from 93.5% to 97.5%. This means the estimation errors are insignificant at the 10% level of significance at all points of the estimated series. Figure presents the SF estimated intercept in comparison to the truth from the 19th sample. Simulation : Measurementerrors-TrueDGPistheUCmodel The hyper-parameters of the UC in this experiment are set to: v = ; = ; µ y0 µ x0 = 0 5 Note that these covariance matrices are chosen to be general in the sense that they are not proportional and the measurement error in the input series exists, vx =36= 0. With this specification, the true structural parameters in the function y t = t + x t + t are = cov / x =1.5 ; t = µ yt 1 1.5µ xt 1 We are going to verify that OLS applied to the first differences fails to estimate the true slope. In contrast, SF is still able to perform this task. Furthermore, we predict that OLS in the differenced series actually returns defined in (1). In our DGP, the actual value of is vy = vx v + y x vx + x = =0.94 Results of Simulation It is strikingly clear from the kernel density plots in Figure 3 that OLS estimates are well below the true 15

16 . Instead, they are centered around a value just below 1. The null hypothesis H o : =0.94 tested for the OLS estimates is not rejected in 9% of the replications, indicating that the parameter that OLS estimates is not significantly different from Hence, is what OLS estimator returns, which is clearly not the true structural parameter of interest. The plot also shows that the SF estimates looks normally distributed, centered around the value 1.5, the true value of. The rate at which the confidence intervals of SF estimates includes 1.5 is 19/00 = 96%. We therefore can be confident that the SF estimator can estimate the true at the 5% level of significance. Figure 3: Distributions of ˆOLS and ˆSF Figure 4: ˆ t,sf from the 19th sample We also construct the pointwise confidence intervals for the estimation errors of the estimated time-varying intercept. The coverage of those confidence intervals over zero at all points in the estimated series ranges from 93.5% to 97.5%, meaning the estimation errors are insignificant at the 10% level of significance at all points in the series. As shown in Figure 4, the estimated intercept tracks its true series well. Simulation 3: Measurement errors - True DGP is the structural model We depart from the UC framework to a different DGP as follows. v yt v xt y t = µ yt + v yt ; x t = µ xt + v xt ; 0 N ; 0 3 µ yt = t + µ xt ; (45) = 1.5 t = t 1 +,t ; 0 = 10;,t N(0, 0.5 ) µ xt = µ xt 1 + xt ; µ x0 = 5; xt N(0,.5 ) 0, The x t and y t series are measured with errors, v yt v xt which are jointly normally distributed. The true relationship is expressed in terms of the underlying correct measures, µ yt and µ xt, in which the slope is constant whereas the intercept varies according to a random walk with drift process. Finally, µ xt is a random walk process with a normally distributed random noise, xt, which has variance of.5. By (45) we have 16

17 µ xt = t 1 N(µ xt 1,.5 ) t = t 1 N( t 1, 0.5 ) Because µ xt and t are independent, µ yt = t 1 = t + µ xt = t 1 N( t 1 + µ xt 1, ). Since µ yt 1 = t 1 + µ xt 1, we can write the marginal distribution of µ yt (conditional on past information) as µ yt = t 1 N( µ yt 1, ) Although the marginal distributions of µ yt and µ xt are both normal, this is not sufficient for their joint distribution to be a bivariate normal distribution. Therefore, without further restrictions, the bivariate normal distribution is not enforced by this GDP. Using this GDP, we can assess the standard Kalman filter (with MLE for the estimation of the hyper-parameters) performance in estimating the constant and time-varying t when multivariate normality does not necessarily hold. Results of Simulation 3 Figure 5 represents the distributions of the SF and OLS estimates of. Clearly, OLS estimates are below the true value, =1.5, whereas the distribution of SF estimates are symmetric, centered around this true value. It is also calculated that the cover rate of confidence intervals constructed from the SF estimates of the UC model over this true value is 96.3%. In contrast, no confidence intervals constructed from OLS estimates includes the true value. Figure 5: Distributions of ˆOLS and ˆSF Figure 6: ˆ t,sf from the 19th sample Moreover, OLS cannot be used to estimate the time-varying parameter t.infigure6weprovideanexample of the estimates of t obtained from the UC representation for the 1st replication in our experiment. 6. Empirical example In this section we apply our approach to the estimation of the model by Diewert and Fox (008) to assess the possible presence of measurement error raised in their paper. 17

18 6..1 Some background on the D&F s paper Assuming cost reducing technical change and constant markup factors within each period across commodities, D&F derive the exact relationship between the growth rates of the input and output. X t = + Y t (46) X t and Y t are Törnquist indexes of the growth in inputs and outputs at period t with base period being t 1, i.e., X t =lnq t (w t 1,w t,x t 1,x t ) and Y t =lnq t (p t 1,p t,y t 1,y t ) where y t and x t are the output and input vectors and w t and p t are the output and input prices, respectively. is the returns to scale parameter and is a technical change parameter, defined as t t 1 where t is the period t productivity parameter 1. The paper acknowledges that in empirical applications this equation may only hold approximately. In order to bridge the theory and practice, it is necessary to allow for a stochastic element in the variables in the form of measurement error. Although acknowledging the measurement error problem and describing three approaches to account for measurement errors, D&F s empirical implementation is using OLS, assuming no measurement errors in X t or Y t. In our model measurement errors are allowed to be present in the observed variables X t and Y t. These may arise from the compilation of the original firm level data, and/or the aggregation methodology of statistical agencies. Due to these reasons, it is also possible that some correlation might be presented between the measurement errors of the data for inputs and outputs for each industry. D&F also raise the issue that it is unclear to practitioners whether one is to regress X t on Y t or Y t on X t when confronted with an exact relationship as (46) because that equation is equivalent to Y t = / +(1/ )X t (47) It is argued that firms in the same industry experience much the same input price growth but the rates of output growth differ systematically across firms, with high productivity growth firms growing faster than low productivity growth firms. With this observation, D&F suggest it is usually better to run the regressions where output growth is the dependent variable. We will follow this specification in our empirical exercise. Let = / and = 1/ ; thus, >0 indicates positive technological change, and >1 implies increasing returns to scale. D&F also suggest converting into a random variable, t = + t,inordertoreflectthefactthattechnical change does not take place in a perfectly smooth manner; new discoveries are often unexpected and thus lead to productivity shocks. Essentially, this implies that the technology change parameter t,andthus, t (= t / ), are random parameters and the period t productivity parameter t is a random walk with a drift parameter. Using the UC framework, we are able to allow t to be time-varying; however, we do not make a priori assumptions regarding the dynamics of t and t. We fit two forms of the UC model, a local level model, which will imply t is random walk process, (see (14)), and a cycle plus trend model, which will imply the technological change also has some cyclical characteristics (see (35)). Through a statistical model selection process, we let the data fit t without imposing strong assumption about its stochastic property. 6.. Data The data set used by D&F consist of the output, inputs and multifactor productivity measures for eighteen industries of the U.S economy. We have to construct Törnquist indexes X t and Y t as described in D&F before using them for our estimation; this is detailed in Appendix B. There are some small differences between the data used in D&F s paper and the data we use here. First, the data available covered the period of 1949 to 001 (one more year from that used by D&F). Second, data for three years from 1950 to 195 are missing. Therefore, the data 1 t is originally used in Diewert and Fox (008) s paper, however, to keep the notation consistent with that in this paper and avoid confusion with our notation for the disturbances of the cycle, we use t. 18

19 Figure 7: Törnquist indexes of the input and output of the U.S aggregate manufacturing sector, Figure 8: Autocorrelation of two indexes we use for our estimation is for the period Despite these facts, the results we obtain by replicating D&F s method on this data set are not very different from D&F s estimates, suggesting the two data sets are comparable. We choose to implement our proposed approach for the aggregate US manufacturing sector and for four industries, Textile Mill Production (SIC), Apparel & Related Production (SIC3), Industrial and Commercial Machinery And Computer Equipment (SIC35), Electrical and Electronic Equipment (SIC36). These four industries are chosen because D&F s analysis suggests that they have different characteristics from those of other industries. They find that there is not enough evidence for the SIC and SIC3 industries to reject the constant returns to scale hypothesis, while increasing returns to scale is a general feature of most industries. Most of the growth is therefore attributed to increasing returns to scale and not to technological progress. The exceptions are SIC35 and SIC36, the two industries that exhibit both increasing returns to scale and positive technological progress. We apply our estimation approach in order to compare our findings with D&F s results Estimation and Results This empirical analysis consists of two parts. First we carry out model selection to determine whether the UC model should be a local level (random walk) or a trend with cycle model. Second, the parameters of the structural model (46) are obtained by estimating the UC model using STAMP. In the following, we provide details of the model selection and estimation using the aggregate manufacturing sector as an example. The same process is applied to the data for the four industries. Model selection We first have a look at the characteristic of the inputs (X t )andoutputs(y t ) growth indexes. In Figure 7 are the plots of the levels of these series that enter the model. They suggest little evidence of positive trend. This is reasonable because by construction they are essentially the first differenced series of the logs of input and output variables (see Appendix B). The data plot suggests some time variation in the relationship between the two series, i.e., from the mid-80s onwards, the two series have smaller variation; however, their gap is larger than it was before the mid-80s. This feature of the data points to the appropriateness of modeling the relationship using time-varying parameters. The correlograms show that although the individual autocorrelations are quite small, both series exhibit some evidence of cyclicality and thus allowing for it might increase the model s fit. SIC stands for Standard Industrial Classification 19

20 We proceed with the estimation of both models, a simple local level model and a cycle plus trend model, and present the summary outputs below. Local level Prediction error variance (PEV)= ; [AIC Yt ; AIC Xt ]=[ 6.183; 6.8]; [BIC Yt ; BIC Xt ]=[ 6.105; 6.74]; Rd =[0.44; 0.40] = ; =0; v = ; v =0.916; There is evidence of significant autocorrelation of order and 3 in the residual series in both X t and Y t (at 10% level of significance). Cycle plus trend model Prediction error variance (PEV)= [AIC Yt ; AIC Xt ]=[ 6.188; ]; [BIC Yt ; BIC Xt ]=[ ; 6.796]; Rd =[0.46; 0.43] = ; =0; v = ; v = 1; c = ; c =0.845, There is no evidence of significant autocorrelation in the residual series in both X t and Y t (at 1% level of significance). All goodness of fit measures are in favor of the cycle plus trend model although the differences between measures of fit for the two models are relatively small. However, the autocorrelation of order and 3 in the residual series of the local level model suggests not all the dynamics have been fit and this is support for the cycle plus trend model. Also, only allowing for trend and irregular components, the local level model attributes most of variations to the irregular components because the trend in X t and Y t is fixed (which is also indicated by the zero variance matrix of level disturbances, ). This would imply there is no relationship between the two index series, which is counter-intuitive. Therefore, we choose cycle plus trend model in order to better extract the underlying relationship between the two index series. Figure 9 presents the estimated components of each series in the model, X t and Y t. Parameters estimation From the estimated hyper-parameters in and c,andthestatevector,(µ yt,µ xt ) 0 and (c yt,c xt ) 0,wecan estimate the structural parameter as defined in (34) and (35). We repeat this process for four sets of (X t,y t ) for the four remaining industries. The graphs with the analysis of components of each series are provided in Appendix C. The hyper-parameters and the structural parameters estimated through the UC framework are reported in Table 1. We also obtain the OLS estimates using our data and report them together with the OLS estimates in D&F s paper (the last two rows). The small difference between them indicates that the difference in two data sets is negligible as expected. Rows two to four contain the estimates of the covariance matrix of the signals of the trends, ( Yt, Xt ). The small variances of the signals indicate that almost all series either have very smooth trends or stay unchanged over the period. Also, for each industry, the trends in the input and output growth indexes are strongly correlated. Notably, the analysis reveals a negative correlation between the trends in the input growth and output growth for the industry SIC35. Referring to Figure 13 in Appendix C, we can see that the underlying trends of X t and Y t only diverged in opposite directions after the early 1980s. The parameters of the cycles in rows five to seven indicate that cycles appear to be important 3 for a multivariate model, STAMP calculates and reports these statistics for each series in (Y t,x t) separately. 4 Rd is the measure of goodness of fit based on first differences, which is more suitable for the data that shows trend movements (Koopman et al., 1995). In our data, some sectors/industries show significant trends, and some do not. However, to have a consistent comparison criterion, we use Rd. 0

21 Figure 9: Level, cycle and irregular components in X t and Y t contributors to the total variation of the observed series, which supports our choice of the model that allows for cyclicality. From rows eight to ten, it is evident that measurement errors are important components of the observed series. Their contribution to the variation of the observed series are comparable to that of the cycles. Importantly, there is strong evidence of a correlation between the measurement errors in X t and Y t. As Proposition indicates, this correlation leads to a bias in the OLS estimator in addition to the bias caused by the measurement errors in X t.however,asbuonaccorsi(010)remarks,theattenuationbiasisonlyguaranteed if the measurement errors in X t and Y t are uncorrelated; when this is not the case, the direction of bias of the OLS estimator is ambiguous. This might explain the different signs of the difference between ˆSF and ˆOLS across sector/industries. In particular, our results agree with D&F s results on the increasing returns to scale of the aggregate manufacturing sector and the industries SIC35 and SIC36. In contrast, while D&F indicates SIC and SIC3 have not been able to exploit scale economies, our results indicate these two industries have either constant or increasing returns to scale. We scale the time-varying intercepts, t,byˆ to obtain the technical change parameters t (since t = t / ) and graph them in Figure 10. D&F find that the estimates of technological process are very modest, and most are insignificant different from zero. The highest significant estimate D&F find is.% technical progress for Textile Mills Productions (SIC). For the aggregate Manufacturing sector, the null hypothesis of no technical progress cannot be rejected. By using the UC framework, we are able to capture time movement of the technological change parameter exhibited for different industries over the period Wefindthatourresultsareonlyconsistent with theirs for the period , that is that technical progress are quite minimal, the highest is about.% for the aggregate manufacturing sector in 1970 and about % for Textile Mills Production. Technical change 5 the model uses one lag and thus the year 1953 cannot be computed, see (35). 1

22 Aggregate Text Mill Prod. (SIC) Apparel & Related Prod. (SIC3) Ind. Machinery &Comp. Eq. (SIC35) Electrical &Electr. Eq. (SIC36) ˆ y ˆ x ˆ ˆcy ˆcx ˆ c ˆvy ˆvx ˆ v ˆSF ˆOLS ˆD&F Table 1: Hyper-parameters of the cycle plus trend model, and the estimates of returns to scale for the aggregate US manufacturing sector and four industries parameters in SIC and SIC3 industries seem to be stable during the period, although some cyclicality is quite clear. Moreover, our model is also able to pick up a large drop in technical progress in Manufacturing sector in 1974, possibly due to the oil crisis. Since the 1980s, we observe a divergence between our results and D&F s. Industrial Machinery & Computer Equipment (SIC35) and Electric & Electrical Equipment (SIC36) exhibit a significant change in technology from % to 6% from 1980 to 001. This time-variation, although well expected, can not be fitted by D&F s method, which only allows for constant technological change. Figure 10: Time-varying technical change of the aggregate manufacturing sector and 4 industries in the US,

23 7 The case where the slope is time-varying - preliminary proposal One of the advantages of the UC representation of the structural relationship is the possibility of modeling a timevarying slope, t. AsreviewedinSection3,Harvey(013)introducesthetimevariationin t through t because t = t y / x. This section proposes this extension in the presence of measurement errors and an estimation strategy for this model specification. Simulations and empirical work of this part will be included in the next version of the paper. We assume that the observed data follow the simple bivariate local level model as in (11) - (1), except the covariance matrix of the signals ( yt, xt ) is time varying, i.e., t = y t y x t y x x For simplicity, we assume for now that the measurement errors are not correlated, i.e., v = vy 0 0 vx This assumption can be relaxed later. The problem now is that we have to estimate two time-varying processes: the underlying series vector (µ yt,µ xt ) and the time-varying correlation t.when t is time-varying, the standard Kalman filter and Maximum Likelihood estimation outlined in Section 5 are no longer applicable because the hyperparameters are the unknown time-varying series. The DCS approach that Harvey (013) proposes to model the time-varying correlation is for series that are jointly normally distributed with constant means and no measurement errors are present. Therefore, this method can not be applied immediately either. We propose a two-step estimation procedure. First, we model the dynamic of the correlation t through that of 4t, which is the correlation between the first differences 4y t and 4x t. This transformation eliminates the timevarying (µ yt,µ xt ) and allows us to concentrate on the time-varying correlation. We will adopt the DCS approach that Harvey (013) proposed for its flexibility (for the full treatment of time-varying correlation and the class of DCS models, see Creal et al. (011, 008); Harvey (013)). As shown shortly, the estimation of the time-varying correlation will call for a simulated likelihood estimation because the likelihood function does not have closed form. In step two, the standard Kalman Filter can be applied to the observed (y t,x t ) since the time-varying component of the system matrices are now known from step 1. Once all hyper-parameters and the state vector are available, the parameters of the structural model are given by (13)-(14). Step1 Modeling t through 4t We transform the UC model in (11)-(1) into the differenced form.. 4y t 4x t = yt xt + v yt v xt v yt 1 v xt 1 (48) We first note that ( vy, vx, y, x ) can be estimated by the method of moments using the equalities (18)-(19): ˆ(0) = ˆ +ˆ v = 1 TX 1 4y h i t 4y T 1 t 4x t 4x t=1 t ˆ(1) = ˆ v = 1 TX 1 4y h i t 4y T t 1 4x t 1 4x t t= (49) (50) t has a time-varying covariance, t y x ;therefore,themethodofmomentscannotprovideanunbiasedand consistent estimate of t.however,wecanstillobtainunbiasedandconsistentestimatesof vy, vx, y, x. 3

24 Let = t 1 = {4y t 1, 4x t 1 }, from (48), we can write the distribution for 4y t and 4x t conditional on past information = t 1 as: 4y t = t 1 N 4x t v yt 1 v xt 1 ; y + vy t y x t y x x + vx Let 4y = y + vy, 4x = x + vx and cov 4 = t y x.standardize4y t by 4y and 4x t by 4x to obtain ỹ t and x t,wehave: ỹ t x t = t 1 N ṽ yt 1 ṽ xt 1 1 4t ; 4t 1 where ṽ yt 1 = v yt 1 / 4y ; ṽ xt 1 = v xt 1 / 4x and 4t = t y x / 4y 4x. The link between t and 4t suggests that we can model t through 4t. The literature on modeling time-varying correlation offers two main approaches. The first is the parameter-driven class of models in which the time-varying parameter is driven by its lag values and an exogenous random shock. Although this exogenous random shock offers some flexibility in modeling, its consequence is that the likelihood function is not available in a closed form, which often calls for computationally intensive Bayesian estimation and inference techniques. The second is data-driven models in which the dynamic of the time-varying parameter is driven by some function of the observed data. The DCS approach proposed by Harvey is one of such models. See Harvey (013); Creal et al. (008, 011) for a complete literature review. In DCS model, instead of relying on the second moment of the observed data as commonly used in practice, the dynamic of the correlation is driven by the score function, which summarizes the characteristics of the whole distribution of the data. In our case, the likelihood function is unavailable in closed form due to the presence of unobserved measurement errors. However, given that we can approximate the likelihood function by a simulated one, and that the score function can be derived from the simulated likelihood function, the estimation of the time-varying correlation is relatively simple. For this reason, we pursue this approach to model the correlation. We first transform 4t by the Fisher transformation in order to ensure 4t [ 1, 1]. (51) (5) 4t = exp( t) 1 exp( t )+1 As mentioned before, the essence of the DCS is that it assumes that the scaled score function. (53) t follows an autoregressive process driven by t = + t 1 + appleu t 1 (54) The scaled score function, µ t, is the first derivative of the likelihood function of the distribution from which the observed data is generated, i.e., the distribution in (5), scaled by the information matrix. The period t log-likelihood function based on (5) is given by: where =(, 1 lnf(ỹ t, x t, = t 1, ) = ln ln(1 4t) 1 (1 4t ) (ỹ t +ṽ yt 1 ) +( x t +ṽ xt 1 ) 4t (ỹ t +ṽ yt 1 )( x t +ṽ xt 1 ) (55), apple). Asdiscussedbefore,thislikelihoodfunctionisnotavailableinexactformbecause(ṽ yt, ṽ xt ) is not observable. Therefore, for the maximization process, we require a simulated likelihood function. Let (ṽ s yt, ṽ s xt) be 4

25 one of the S draws from the normal distribution with mean zero and covariance matrix: ṽ = The likelihood function (55) can be approximated by: lnf(ỹ t, x t = t 1, ) 1 S SX s=1 s=1 The scaled score based on this approximated likelihood function is vy / 4y 0 0 vx / 4x SX lnf(ỹ t, x t ṽ yt 1 =ṽyt s 1, ṽ xt 1 =ṽxt s 1, = t 1, ) (56). where u t = I 1 t t 1 S t t 1 S t t 1 t, x t, = t 1, ) 1 S I t t 1 t SX s=1 s=1 1 g t ḡt g t +(ỹ t +ṽ s yt 1)( x t +ṽ s xt 1)(g t 1) (ỹt +ṽ yt 1 ) +( x t +ṽ xt 1 ) g t ḡ t where ḡ t =(exp( t )+exp( t))/ and g t =(exp( t ) exp( t))/. Forderivations,seeAppendixA. The log-likelihood function used to estimate =(,, apple) is: L(4y t, 4x t ) = lnf(ỹ t, x t, t =1,,...,T) TY = ln f(ỹ t, x t = t 1, ).f(y 0,x 0 ) TX t=1 t=1 1 S SX s=1 s=1 SX lnf(ỹ t, x t ṽ yt 1 =ṽyt s 1, ṽ xt 1 =ṽxt s 1, = t 1, )+lnf(y 0,x 0 ) Maximizing this likelihood, we will obtain the set of estimates: =(,, apple), which allows us to calculate ˆ 4t through (54) and the covariance of t through t d y x =ˆ 4tˆ4y ˆ4x (57) Step : Use the Kalman Filter to obtain (µ yt,µ xt ) (see Section 5), replacing v and with the estimated ˆ v and ˆ t : ˆvy 0 ˆ v = and ˆ ˆ y d y = 0 ˆvx t d y x ˆ x t y x If we suspect there is correlation between the measurement errors, the method of moments in Step 1 is still able to estimate this covariance. We can then use this estimate to extract the covariance of the signals from the covariance of the first differences. More specifically, we can implement step 1 as specified above except the covariance of t in (57) is replaced by t d y x =ˆ 4tˆ4y ˆ4x cov(v ˆ yt,v xt ).Stepremainsunchanged. 5

26 8 Conclusions The paper considers the estimation of structural parameters of interest in the presence of measurement errors in all the variables involved in the economic relationship. The approach proposed uses the time series dynamics together with properties of the multivariate normal distribution in order to identify the structural parameters of interest. The approach allows the measurement errors in the variables to be correlated, as well as accommodates easily the presence of trends, cycles and seasonality in the observed data. In the present version of the paper we use a bivariate relationship and leave the extension to more than two variables to the next version of the work. We cover two cases in this paper. In the first the economic relationship has a time-varying intercept and a constant slope parameter. In the second both parameters are time-varying. We provide the estimation approach, simulations and an empirical example for the first case and provide the proposed estimation approach for the second case. Simulations and empirical implementation of the second case are work in progress. The case when only the intercept is time-varying allows for the unobserved components model to be written in a standard state-space form which provides practitioners with a number of alternative estimation packages. We implement the empirical example in this paper using STAMP. Two propositions are proved to show how the proposed unobserved components can identify the structural parameters of interest, and the exact form of the bias in the least squares estimator of the slope parameter as a function of the variances and correlation between measurement errors in the variables. The simulation results indicate that the unobserved components framework can give consistent estimates of the parameters in the presence of measurement errors even when the measurement errors are correlated and the true data generation process for the structural model does not necessarily follow a multivariate normal distribution. The empirical example is that from Diewert and Fox (008) who derive a number of theoretical results on estimating returns to scale and technical progress when there are multiple outputs and inputs. They raise the likelihood of significant measurement errors in the input and output indexes used to estimate their theoretical relationship. However, they do not attempt to account for them in the estimation. Our results indicate strong evidence of measurement errors in the aggregate indexes of input and output growth, as well as of a significant correlation between these measurement errors. The general conclusion is consistent with Diewert and Fox (008) s result, in that most of the industries growth has been due to increasing returns to scale and that technological progress is minimal. The exceptions are for industries in Textile Mill Production (SIC) and Apparel and Related Production (SIC3), where we found constant and increasing returns to scale, respectively, instead of decreasing returns to scale as claimed by Diewert and Fox (008). The degree of increasing returns to scale differs in some cases from those found by Diewert and Fox (008). We conclude this ambiguity in the direction of OLS bias is due to a strong correlation between the measurement errors of the input and output indexes used in the estimation. By allowing for the time variation in the intercept, and thus, the technological change parameter, we find evidence of variation in the technological progress across time and industries. In particular, Textile Mill Production (SIC) industries exhibit quite clear cyclical patterns in technological change. Industrial and Commercial Machinery And Computer Equipment industries (SIC35), and Electrical and Electronic Equipment industries (SIC36) show a significant increase in the technological progress since the 1980s. These are expected results; however, not obtainable if we use least squares to estimate the structural model as Diewert and Fox (008) did. References Anderson, B. D. O. and Deistler, M. (1984). Identifiability in dynamic errors-in-variables models. Journal of Time Series Analysis, 1(4):

27 Anderson, T. (1976). Estimation of linear fufunction relationships: Approximate distribution and connections with simultaneous equations in econometrics. Journal of Royal Statistical Society, Series B 38, pages1 36. BLS (015). Superceded historical sic measures for manufacturing sectors and -digit sic manufacturing industries, Buonaccorsi, J. P. (010). Measurement error: Models, Methods, and Applications. TaylorandFrancis. Creal, D., Koopman, S. J., and Lucas, A. (008). Generalized autoregressive score models with applications. Journal of Applied Econometrics, 8: Creal, D., Koopman, S. J., and Lucas, A. (011). A dynamic multivariate heavy-tailed model for time-varying volatilities and correlations. Journal of Business and Economic Statistics, 9(4). Devereux, P. J. (001). The cyclicality of real wages wiwith employers-employee matches. Industrial and L, 54(4): Diewert, W. E. and Fox, K. J. (008). On the estimation of returns to scale, technical progress and monopolistic markups. Journal of Econometrics, 145(1 ): Theuseofeconometricsininformingpublicpolicymakers. Durbin, J. and Koopman, S. J. (01). Time Series Analysis by State Space Methods. OxfordStatisticalscience series. Falk, B. and Lee, B.-S. (1990). Time-series implications of friedman s permanent income hypothesis. Journal of Mon, 6: Fuller, W. A. (1987). Measurement Error Models. WileySeriesinProbabilityandMathematicalStatistics. Granger, C. W. J. (1986). Developments in the study of cointegrated economic variables. Oxford Bulletin of Economics and Statistics, 48:13 8. Greene, W. (01). Econometric Analysis. Boston. Harvey, A. C. (1989). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. Harvey, A. C. (011). Modelling the phillips curve with unobserved components. Applied financial economics, 1:7 17. Harvey, A. C. (013). Dynamic models for volatility and heavy tails with application to Financial and economics time series. Cambridge University Press. Hassler, U. and Kuzin, V. (009). Cointegration analysis under measurement errors. In Measurement Error: Consequences, Applications and Solutions, pages Kapteyn, A. and Wansbeek, T. (1983). Identification in th linear errors in variables model. Econometrica, 51(6). Kendall, S. M. and Stuart, A. (1977). The Advanced theory of statistics, volume Inference and Relationship. Charles Griffin & Company Limited, 4 edition. Klette, T. and Griliches, Z. (1996). The inconsistency of common scale eestimator when output prices are unobserved and endogenous. Journal of applied econometrics, 11: Komumjer, I. and Ng, S. (014). Measurement errors in dynamic models. Econometric Theory. 7

28 Koopman, S. J., Harvey, A. C., Doornik, J. A., and Shephard, N. (1995). STAMP 5.0 Structural Time Series Analyser, Modeller and Predictor. Chapman&Hall. Maravall, A. and Aigner, D. J. (1977). Identification of the dynamic shock-error model : The case of dynamic regression. Latent Variables in Socio-economic models Editted D. J. Aigner and A. S. Goldberger. Marschak, J. and Andrews, William H., J. (1944). Random simultaneous equations and the theory of production. Econometrica, 1(3/4):pp Nowak, E. (1993). The identification of multivariate linear dynamic errors- in- variables models. Journal of Econometrics, 59:13 7. Ornaghi, C. (006). Assessing the effects of measurement errors on the estimation of production functions. Journal of Applied Econometrics, 1(6):pp Phillips, P. C. and Durlauf, S. N. (1986). Multiple time series regression with integrated processes. Review of Economic Studies, 53(4): Reisersol, O. (1950). Identifiability of a linear relation between variables which are subject to erro. Econometrica, 18(4): Sargent, T. J. (1989). Two models of measurements and the investment accelerator. Journal of Politica, 97(): Solo, V. (1986). Identifiability of time series models with errors in variables. Journal of Applied Probability, 3: Thiele, S. and Harvey, A. C. (013). Time varying regression and changing correlation. in Workshop on Dynamic Models driven by the Score of Predictive Likelihoods, organized by VU University Amsterdam, Tinbergen Institute, and University of Cambridge. Van Beveren, I. (01). Total factor productivity estimation: A practical review. Journal of Economic Surveys, 6(1): A Derivation of the conditional scaled score for the simulated likelihood function in Section 7. The score function is given by: S t t 1 t, x t = t 1, t t ( 1 S 1 S SX SX s=1 s=1 The information quantity is given by: SX s=1 s=1 SX lnf(ỹ t, x t ṽ yt 1 =ṽyt s 1, ṽ xt 1 =ṽxt s 1, = t t lnf(ỹ t, x t ṽ s yt, ṽ s xt= t 1, ) (58) 8

29 ln(ỹ t, x t = t 1, ) I t t 1 = E E 1 S 1 S SX s=1 s=1 t s=1 s=1 ln(ỹ t, x t vyt s 1,vxt s 1, = t 1, t ln(ỹ t, x t vyt s 1,vxt s 1, = t 1, E Here, we adapt the derivations of the score and information matrix in Harvey (013, pp 15-16). The first derivative of the period t t, x t ṽ s yt 1, ṽ s xt 1, = t 1, t = ḡt g t +(ỹ t +ṽ s yt 1)( x t +ṽ s xt 1)(g t 1) (ỹt +ṽ yt 1 ) +( x t +ṽ xt 1 ) g t ḡ t (59) The second derivative of the individual likelihood lnf(ỹ t, x t ṽ yt 1, ṽ xt 1, = t 1, t = g t ḡ t g (ỹ t +ṽ yt 1 ) (g t +ḡ t )+(ỹ t +ṽ yt 1 )( x t +ṽ xt 1 )4g t ḡ t ( x t +ṽ xt 1 ) (g t +ḡ t ) So the score function is defined as (58) and the element of the sum is given by (59). Using the identity g ḡt =1, t =ḡ t /g lnf(ỹ t, x t ṽyt s 1, ṽxt s 1, = t 1, E = 1 +gt +ḡt 4ḡt = (g ḡ )= g t 1 g t Therefore, the information matrix (scalar) is I t t 1 1 S SX SX ( s=1 s=1 1 ) g t 1 g t B Data preparation X t, the Törnquist input quantity index, is an aggregate of five inputs, including capital, labor, energy, materials and purchased business services inputs. X t = 1 5X (s t n + s t n 1 )(lnx t n lnx t n 1 )= 1 n=1 5X (s t n + s t n 1 )ln(x t n/x t 1 where s t n is the cost share of the input n at time t, s t n = cost of x t n/total cost t. The cost of each input and the total cost of production are available in the data set. However, the data set does not contain the level of each input used. Only the annual percentage change and the index for each input (the base year is t = 1996) arepresented. Using these two series, we can calculate x t n/x t 1 n in two ways: n=1 n ) x t n x t 1 n = xtn/x1996 n x t n 1 /x 1996 n = index year t index year t 1 9

30 or x t n x t n 1 = x t n x t 1 n x t 1 n %annual change +1 = In relation to the output quantity index, Y t can be calculated as where Y t = 1 MX ( m t + m t 1 )(lnym t lnym t 1 )= 1 m=1 MX ( m t + m t 1 )ln(ym/y t t 1 t n is the share of output m. Output in these data is based on the deflated added-value of production. In this sense, these output data are aggregate data. So in applying the Törnquist formula above, we let m =1and t thus, m = m t 1 =1. ym/y t m t 1 is calculated in the same way as x t n/x t n 1 described above. C m=1 Component Estimates for the four industries m ) Figure 11: Components of Y t and X t for SIC 30

31 Figure 1: Components of Y t and X t for SIC3 31

32 Figure 13: Components of Y t and X t for SIC35 3