Fresh perspectives on unobservable variables: Data decomposition of the Kalman smoother

Transcription

1 Fresh perspectives on unobservable variables: Data decomposition of the Kalman smoother AN 2013/09 Nicholas Sander December 2013 Reserve Bank of New Zealand Analytical Note series ISSN Reserve Bank of New Zealand PO Box 2498 Wellington NEW ZEALAND The Analytical Note series encompasses a range of types of background papers prepared by Reserve Bank staff. Unless otherwise stated, views expressed are those of the authors, and do not necessarily represent the views of the Reserve Bank.

2 Reserve Bank of New Zealand Analytical Note Series 2 Non-technical summary Macroeconomic analysis, including that undertaken by the Reserve Bank, makes extensive use of economic concepts for which no observed data exist. The output gap is one well-known example. The gap must be estimated using various models and indicator variables. Statistical estimates of core inflation, such as the sectoral factor model (Kirker, 2010), are another example. In working with estimates of unobserved variables, it can be difficult to articulate the contribution of each piece of observed data to the estimate of an unobservable variable. The data decomposition tool outlined in this note does exactly that. The tool is applied to two types of unobserved variables. The first is the sectoral factor model estimate of core inflation. That model takes 96 component series of the CPI and uses statistical techniques to identify the co-movement among those series (separate tradable and non-tradable factors). The data decomposition tool enables us to identify which component series have had the largest influence, on average through time or in any particular quarter, on the estimate of core inflation. It turns out that over the 20 year history of the series three components (with a cumulative weight of 1.04 percent of the CPI) contributed around 30 percent of the variability in core inflation. Tests suggest that the methodology is robust to plausible measurement errors in these, and other, individual price series. The second application is to a small structural model of the New Zealand economy. In modern structural models, variables are assumed to be driven by structural shocks (such as a monetary policy shock ) which are also not directly observable. We use the data decomposition technique to illustrate the impact on the estimated shocks of adding an additional quarter of actual data. We can t directly observe the behaviour of economic agents, so the data decomposition describes the process macroeconomists go through to infer from new data which shocks must have occurred in the economy.

3 Reserve Bank of New Zealand Analytical Note Series 3 1 Introduction A common method for analyzing the state of the economy is to estimate unobserved component models. These models describe the behaviour of observable data in terms of underlying economic concepts that are unobservable. Although some of these models impose very strict structures that give a unique value for an unobservable variable, most impose a lighter structure where many possible values of the unobservable variable are consistent with the observed data. 1 The unobserved components in the models are estimated using filtering techniques such as Kalman smoothing or particle filtering. 2 One key advantage of the filtering techniques is that they can be applied to a wide range of models without requiring any adjustment to the filtering algorithm. However, the generalized nature of these filters can obscure how observable data are processed for a specific model. In cases where these filters are given a single observable variable, the unobservable estimates are typically determined by decomposing the observable data series by frequency (i.e. cycles in the data). Because of this, most applications of filtering with one observable variable involve estimating low-frequency trends and high-frequency cycles in macroeconomic data. 3 However, for models with multiple observable variables, it is not immediately clear how the observable data is processed to produce estimates of unobservable variables. Understanding how these filters infer from observable data the likely values of unobservable variables could yield useful insights into unobservable estimates and model behaviour. This note develops a tool the data decomposition of the type discussed in Andrle (2013a and 2013b) that is designed to separate the contributions of the various observable variables to the Kalman smoother s estimates of unobservable variables. 4 Because of the general nature of the Kalman smoother algorithm, this decomposition can be applied to any linear model that can be written in state space form. 1 The HP filter introduced in Hodrick and Prescott (1997) is a commonly used example of a model where the trend is unique given the data (and smoothing parameter). DSGE models with unobservable marginal costs (such as Smets and Wouters (2004)) and dynamic factor models as in Cristadoro et al. (2005)) are two examples of models without unique unobservables. 2 Simon (2006) and Durbin and Koopman (2012) both provide comprehensive coverage of both of these techniques. 3 Some examples are: Beveridge and Nelson (1981),Harvey and Jaeger (1993), King and Rebelo (1993),Cogley and Nason (1995),Harvey (1985), Clark (1987) and Watson (1986) 4 The concept of a data decomposition was first proposed by Andrle (2013a) and applied in Andrle (2013b) to decompose various output gaps into observable data. While these papers present methods to infer the contributions of observable data by exploiting the linearity of the model, this article uses the weights derived in Koopman and Harvey (2003) for the Kalman smoother. Using explicit weights has the computational advantage of requiring the Kalman smoother be run only once and provides additional information regarding how these weights change throughout the sample.

4 Reserve Bank of New Zealand Analytical Note Series 4 Through two examples of policy-relevant models, this note demonstrates that the data decomposition can provide useful insights to policymakers by analyzing unobserved variables and model-based forecasts. This article is structured as follows: section 2 discusses state space models and the Kalman smoothing procedure an understanding of these techniques is useful in understanding intuitively what the data decomposition does, section 3 applies the data decomposition to the two unobservable components models mentioned above and section 4 concludes. An algorithm detailing how to compute the outputs required for the data decomposition is provided in the appendix. 2 Unobservable Components Models and the Kalman Smoother Estimating unobservable variables is an important process in macroeconomic modelling. One of the more common tools used to estimate unobserved variables in these models is the Kalman smoother. The Kalman smoother has the advantage of being relatively simple to implement while being applicable to a large class of models. The disadvantage of this flexibility is that the precise manner in which the Kalman smoother estimates unobservable variables for specific models can be lost. The data decomposition tool can be used to address this. This section provides an overview of how the Kalman smoother works (with technical details provided in Appendix A). Understanding, in non-technical terms, how the Kalman smoother works can provide some insights into how the data decomposition works and how its output should be interpreted. 2.1 State Space Unobserved components models are commonly written in a State Space Form. This structure essentially distinguishes between the fundamental concepts that describe the evolution of a system and the data that are observed by the econometrician or statistician. The mathematical notation for the state space of a linear and Gaussian 5 unobserved components model is given below: 5 Non-linear models with non Gaussian errors can also be written in state space form, but they are not discussed here.

5 Reserve Bank of New Zealand Analytical Note Series 5 X t = AX t 1 + Rɛ t (1) Y t = BX t + e t (2) ɛ t N(0, Q) e t N(0, H) The notation X, Y, ɛ and e each refer to groups of variables and the remaining terms A, B, R, Q and H represent groups of parameters that link the variables together. 6 Equation 1 describes the evolution of the system and is often referred to as the transition equation. It describes how a group of n possibly unobservable variables (denoted by X) evolve between discrete time periods (from t 1 to t). 7 The variables X are driven by unobserved inputs ɛ. Because the inputs ɛ are unobservable and their evolution unmodelled we describe them as being random in nature and arising from a normal distribution. 8 It is common to refer to these inputs as shocks or structural shocks. The evolution of the variables X over time is determined by the parameter matrix A. The response of variables X to various structural shocks ɛ is governed by the parameter matrix R. Equation 2 describes how the true drivers of the system X generate the observable data Y and is called the measurement equation. 9 The state space form allows for there to be measurement errors e that reflect the imperfect measurement of the data Y. These measurement errors are also assumed to originate from a normal distribution. The observable data Y is linked to the 6 X t, Y t, ɛ t and e t are of size n 1, m 1, n ɛ 1 and n e 1 respectively. In addition, A, B, R, Q and H are of size n n, m n, n n ɛ, n ɛ n ɛ and m m respectively. Note that for the Kalman smoother to estimate properly, n ɛ must be greater than or equal to the number of observable variables m 7 Many models contain structures that impose dependence between variables spanning multiple periods - rather than just one as indicated in the above equation. These models can be incorporated into the above structure by redefining the lag of a variable as an additional ( ) variable [ in X. ] ( Consider ) for example the ( following ) system: d t [ = ρ 1d t 1 ] +ρ 2d t 2. dt ρ1 ρ 2 dt 1 dt ρ1 ρ 2 This can be rewritten as: =. Defining X d t d t = and A = gives the t 2 d t following state space system: X t = AX t 1. This can be done for any multiple lag model. 8 The assumption of normality is not necessary to execute the Kalman smoother algorithm, however, the estimates from the Kalman smoother will not be optimal without Gaussian errors. 9 This structure should not be viewed as forbidding observable data representing concepts of economic relevance. If there is data available that represents an underlying driver of economic fluctuations such as government spending, then the state space form can be re-written as follows: [ ] ( ) [ ] ( ) [ ] Xt A Xt 1 R ɛt = + g t [ Yt G t 0 ] ( = g t 1 B ) [ ] [ ] Xt et + 0 In this model, G, g and ɛ g each represent the same concept: government spending. By including the variable ɛ g government spending represents an unmodelled input for which there is data. g t ɛ g,t

6 Reserve Bank of New Zealand Analytical Note Series 6 unobservable data X by the parameter matrix B. Each observable data series is allowed to be related to multiple variables from X, or just one if desired. The parameter matrices Q and H represent the variance parameters for the stochastic inputs ɛ and the measurement errors e respectively. Stochastic inputs occurring in the same time period are allowed to be correlated with each other as are measurement errors however, the measurement errors are assumed to be uncorrelated with the stochastic inputs. As an example of how to write a model in state space form, consider a simplified version of Harvey (1985) that calculates the output gap: y t = t t + c t + e y t t t = t t 1 + ɛ t t c t = 0.6c t 1 + ɛ c t e y t N(0, σ e ) ɛ t t N(0, σ ɛ t) ɛ c t N(0, σ ɛ c) This is a model where there are three components: GDP (y t ), which is observed with a measurement error e y t ; potential or trend GDP (t t), an unobservable variable; and the output gap (c t ) the unobservable object of interest. There exist some structural shocks ɛ t t and ɛ c t which act as drivers of potential GDP and the output gap; and (as mentioned above) are assumed to be random normally distributed variables. In this model movements in potential GDP are defined as causing movements in GDP that are permanent absent any shock ɛ t t. Movements in the output gap by contrast are defined as causing movements in GDP that eventually decay to 0 even in the absence of any shock ɛ c t. The rate at which the output gap decays to zero is set to 0.6 per quarter for convenience. In other words, if the output gap were 1% of GDP this quarter, next quarter assuming no new shocks it would be 0.6% of GDP. The measurement error is defined as a movement in GDP that completely vanishes in the next period (again, absent any subsequent shock). It is these differences in variable behaviour rather than any relationship to other variables such as inflation that the Kalman smoother will exploit to estimate each unobservable variable.

7 Reserve Bank of New Zealand Analytical Note Series 7 In state space form this model becomes: t t = 1 0 t t 1 + ɛt t c t c t 1 ɛ c t }{{}}{{}}{{}}{{} X t A X t 1 ɛ t [ ] y }{{} t = 1 1 t t + e y t c }{{} Y t t e t }{{} B }{{} X t (Transition Equation) (Measurement Equation) 2.2 The Kalman Smoother The Kalman smoother is a tool that can estimate the unobservable variables in any linear state space model. In this way, the Kalman smoother provides economists with estimates of some of the economic concepts and variables they are most interested in. Because the algorithm for the Kalman smoother can be applied to any model written in state space form, it is not clear how the algorithm produces estimates of unobservable variables for a particular model. The data decomposition is used to understand the output from this estimation procedure in a modelspecific context. However, an understanding of the Kalman smoother procedures (independent from any model) can be useful in interpreting the output from the data decomposition. In the model of potential GDP and the output gap in the previous subsection, there are infinitely many combinations of the unobservable GDP trend, the output gap and measurement errors that could explain the evolution of observable GDP data. Because of this, the Kalman smoother is not able to estimate potential GDP or the output gap with absolute certainty. Under the assumption of normally distributed errors (and a correctly specified model), the Kalman smoother does however, produce the optimal estimate of the statistical distribution of these unobservables given the GDP data supplied. The mean of this (normal) distribution represents the statistically most likely estimate of potential GDP and the output gap. To estimate potential GDP and the output gap over the entire sample efficiently, the Kalman smoother uses all the available data to inform the estimate at each point in time including data occurring both before and after that particular point in time (if available). To illustrate this point, consider using a sample of GDP data from 2000Q1 to 2013Q1 to estimate the value of potential GDP and the output gap as at 2006Q1. The model structure implies that a rise in potential GDP for 2005Q4 will likely be followed by a similar rise in potential GDP for

8 Reserve Bank of New Zealand Analytical Note Series 8 the following period. In addition, a rise in the output gap for 2005Q4 predicts that the output gap over the near future will (absent any surprises) slowly decline back to 0. Therefore, if 2005Q4 GDP data suggests changes in the estimates of potential GDP or the output gap in that quarter, this should also suggest changes to estimates of the values of potential GDP and the output gap for 2006Q1 (and so on). Similarly, data for periods following 2006Q1 are useful in determining potential GDP and the output gap in 2006Q1. If GDP were observed to increase in 2006Q2, it is possible that this increase was due to changes in potential GDP or the output gap in 2006Q1. Data from both before and after 2006Q1 may hold some insights into the unobservable variables for 2006Q1. The manner in which these insights are formed depends on the model structure and the data provided. The Kalman smoother exploits these insights in its estimation procedure. Unlike other estimation procedures such as ordinary least squares, Kalman smoother estimation proceeds recursively as shown in figure 1. That is, the Kalman smoother processes the observable GDP data one period at a time. It moves through the data both from the beginning of the sample to the end (forward) and then, having processed the entire sample, from the end to the beginning (backward). At each part of the estimation, the mean values of the output gap and potential GDP are estimated as well their variances. 10 Iterating forward is referred to as filtering and the addition of iterating backwards after filtering is referred to as smoothing. Figure 1: Kalman smoother recursions 1. Filtering Predict Update time t t+1 t+2 t+3 Update 2. Smoothing The Kalman filter is often described as having two phases: prediction and updating. These operations describe the forward recursive estimation of potential GDP and the output gap at each point in the available sample. In the prediction phase, the previous estimates of the mean 10 Because the estimates of the unobservable variables X t are normally distributed, knowledge of the mean and variance of the unobservable estimates is sufficient to characterize the entire distribution of the estimates.

9 Reserve Bank of New Zealand Analytical Note Series 9 and variance of the output gap and potential GDP are used to forecast the current mean and variance of these unobservable variables. In constructing these forecasts, the unmodelled inputs into the system (ɛ t t and ɛ c t) are assumed to be at their mean value of 0. Using the model structure specified in the Transition Equation above, the prediction for potential GDP will be the same as the previous period and the output gap prediction will be 60% of the previous period s estimate. In the updating phase, the implications of these forecasts on observable GDP are calculated. If these forecasts suggest a value of GDP that is at odds with the observable data, the forecasts of potential GDP and the output gap will be updated to better reflect the observable data. The updated estimates of the mean and variance of the unobservable variables are referred to as filtered estimates. 11 The relative weights applied in the updating phase between new data and the prediction from the model varies between each iteration. These weights make up a matrix referred to as the Kalman gain. The Kalman gain efficiently links the new data to the prediction by accounting for both the uncertainty of the previous estimate of the unobservable variables and the variance of the measurement error. 12 The smaller the measurement errors and the stronger the link between the unobservable variable and the observable data, the greater the weight placed on the new data relative to the prediction. The smaller the variance of the previous estimate, the more weight is placed on the previous prediction relative to the new data. Moving an extra period through the sample introduces a new period of data. The Kalman filter only uses this data to update estimates of unobservable variables occurring in the same time period as the data. Previously occurring estimates of unobservable variables are not updated using the new data until the smoothing stage. This final stage contains a single updating phase which iterates backwards from the end of the sample back to the beginning to incorporate information processed in later iterations of the Kalman filter into earlier estimates of potential GDP and the output gap. If, for example, GDP data for 2006Q1 rose sharply and remained at similar levels for some time, the most likely explanation (according to the model being applied) is that this was due to an increase in potential GDP in 2006Q1. This is because potential GDP is characterized in the model by its strong persistence and this persistence is able to explain the observed GDP data 11 The variance of the unobservables varies over the data set because of the different amounts of information available in forming the estimates of the unobservable variables. As more iterations of the filter take place, each subsequent estimate of the unobservable variables is based on more information. This reduces the estimate s uncertainty. The variance of the unobservable variables therefore, tends to fall further into the sample (and converge toward a positive value). This is not the case after smoothing because each estimate is informed by the entire set of available data rather than simply all current and preceding data. 12 As mentioned in subsection 2.1 the variance of the measurement errors are assumed to be known when running the Kalman smoother.

10 Reserve Bank of New Zealand Analytical Note Series 10 subsequent to 2006Q1. The Kalman smoothing phase will update the estimate of potential GDP in 2006Q1 to reflect that GDP remained elevated in 2006Q2 and beyond. The degree to which the filtered estimates are adjusted in response to data occurring after 2006Q1 depends on both the relevance of the extra data to the 2006Q1 estimates and the confidence around the filtered estimates. If, for example, there was considerable uncertainty regarding the filtered estimate of potential GDP in 2006Q1, then accounting for the higher subsequent GDP data will likely have a large impact on the smoothed estimate of potential GDP in 2006Q1. In summary, the Kalman smoother efficiently uses all the observable data to estimate each unobservable variable at each point in time over the sample. The result is an estimate of the distribution of the unobservable variables that is as precise as statistically possible (given the assumptions mentioned in subsection 2.1). The mean of this distribution represents the most likely estimate of the unobservable variables that generated the observable data. 13 The data decomposition used in the next section, identifies how observable data is processed to arrive at the mean estimates of the model s unobservable variables. In the output gap example mentioned above, this is trivial since there is only one observable variable: GDP. However, in larger models it can be helpful to understand what influence various observed data are having on the unobserved variables being estimated. 2.3 What is the Data Decomposition? The data decomposition splits the estimate of an unobservable variable into contributions from the various observable data. Each observable data series has an effect independent from other observable data because the Kalman smoother s estimate of each unobservable variable is always a linear function of the observable data. Because linear functions can always be separated into independent terms, the effect of each observable variable on the Kalman smoother s estimate of a particular unobservable variable can be separately identified. This process of separating variables within a time period, and then across time as the smoother moves between time periods (both forward and backward), determines the weights on each observable variable used to construct the data decomposition. The technical details of this process and how they can be implemented are included in Appendix A.2. As mentioned above, the estimate of an unobservable variable at a particular point in time is 13 While the mean produced represents the best estimate of the unobservable variables given the available sample, these estimates are most inaccurate near the beginning and end of the sample. This is known as the end-point problem.

11 Reserve Bank of New Zealand Analytical Note Series 11 informed by observable data throughout the entire sample. Therefore, when interpreting the data decomposition, it is important to view the contributions as summarising all data both before and after the quarter being considered. 3 Applications In this section the data decomposition is applied to unobservable estimates originating from two different models. 3.1 Sectoral Factor Model of Core Inflation The sectoral factor model of core inflation as detailed in Kirker (2010) and Price (2013) is a dynamic factor model. Factor models are tools used to summarize the information in large datasets. 14 Essentially, the method is to estimate unobservable factors that represent common variation between all the data series. A factor representing co-movements in prices in the economy is useful to central banks in that it is structured to ignore idiosyncratic movements in specific price indices. These idiosyncratic movements may affect headline CPI inflation, yet likely reflect relative price movements rather than the concept of general price increases (core inflation) that central banks are primarily concerned with. Dynamic factor models include additional structure detailing how the factor evolves over time. The sectoral factor model of core inflation uses estimates of two factors from a data set of 96 disaggregated price indices (components of New Zealand s CPI) to model core inflation: a factor from 56 disaggregated price indices classified as tradable and a second factor from 50 disaggregated price indices classified as non-tradable (10 price indices are classified as a mix of prices in tradable and non-tradable economic sectors). This allows the model to differentiate between price pressures originating domestically and those originating from internationally traded goods and services. These factors are then combined to create the summary estimate of core inflation. 15 Figure 2 shows the tradable factor from the sectoral model against the (standardized 16 ) See Forni and Lippi (2001) and Stock and Watson (2002) for details on factor models and how they summarize economic data. 15 The published measure can be accessed here: 16 A standardized data series is re-scaled so that the series has a mean of 0 and a variance of 1.

12 Reserve Bank of New Zealand Analytical Note Series 12 Figure 2: Tradable factor 5 Tradable data (56 series) 4 Standardised annual percent price change Tradable factor This figure details how the tradable factor from Kirker (2010) (shown in black) fits through all 56 disaggregated price series classified as tradable. Source: Price (2013) Figure 3: Core inflation estimate % Annual Percent Change in NZ CPI - Core CPI

13 Reserve Bank of New Zealand Analytical Note Series 13 series classified as tradable. As the graph shows, the factor tends to run through the areas that are most densely populated with tradable price series thus capturing the co-movement between these series. Although the factors identify the common movements in all the data, some data series have smaller idiosyncratic movements than other variables. These variables are therefore more useful in determining the best estimate of the factor. The data decomposition can identify which data series have had the largest contribution to the factor estimates. 3.7 Figure 4: Data decomposition of core inflation Core Hairdressing - Veterinary services - Household appliance repairs - Vehicle servicing and repairs - Dental Services - Property maintenance services - Ready-to-eat food - Gas - Telecommunication services - Accommodation services - Other Data This figure shows the data decomposition from the Core CPI model in Kirker (2010). THe black line is the overall estimate of core CPI and the coloured bars represent the contributions from the top ten most important data series (by mean absolute error). The grey bars represent all other data. The estimate of core inflation 17 against overall New Zealand CPI inflation is shown in Figure 3 and Figure 4 shows the data decomposition of core inflation. The black line in both figures represents the estimate of core inflation and the coloured bars in Figure 4 show the contributions from the individual price indices. The decomposition is centred around the series mean of 2.2 percent. For simplicity, we show the ten data series (out of 96) that have had the largest con- 17 This estimate of core was produced using data up to June 2013.

14 Reserve Bank of New Zealand Analytical Note Series 14 tributions to the estimate of core inflation over the full period and leave the total contributions from the other series in grey. Overall, the top ten data series have explained about 60 percent of the variation in core CPI and the top three components (with a total CPI weighting 0f only 1.04 percent) have explained approximately 30 percent of the variation. 18 Kirker (2010) noted that the sectoral factor model s estimate of core inflation is largely explained by the non-tradable factor. Thus it is unsurprising that the five most important price series in determining core CPI come from non-tradable subsectors of the economy (these series are hairdressing, veterinary services, household appliance repairs, vehicle servicing repairs and dental services). These series are quite labour-intensive subsectors and labour market pressures are often regarded as a key element in domestic inflation. Other CPI component series which could also represent labour intensive sectors in the economy could have been less useful as indicators of overall labour market pressures than hairdressing or veterinary services price data if these sectors faced fluctuations in their costs (such as raw material costs) that were specific to their industry. 3.4 Figure 5: Effect of modified data on core CPI Effect of Modified Data on Core % Original Estimate of Core CPI - Estimate of Core CPI with Modified Data This figure details how core CPI is revised when hairdressing, veterinary services and appliance repairs data are increased by 1 standard deviation in 2003Q1. The blue line represents the original estimate of core CPI with original data. The black line represents how core CPI is revised when the three most important data series in terms of contributions are increased by 1 standard deviation in 2003Q1. 18 The precise contributions to core inflation have varied over considerably over time. While the data decomposition could be used to analyze this, this is beyond the scope of this note.

15 Reserve Bank of New Zealand Analytical Note Series 15 Figure 6: Data decomposition of the change in core CPI due to adjusted data Core Hairdressing - Veterinary services - Household appliance repairs This figure details how core CPI is revised when hairdressing, veterinary services and appliance repairs data are increased by 1 standard deviation in 2003Q1. The black line represents the total revision to core CPI when all three data series are adjusted and the coloured bars each represent the revisions to core CPI from the adjustment to each individual data series. That hairdressing, veterinary services and household appliance repairs price indices have explained approximately 30 percent of core inflation fluctuations appears to suggest that the estimate of core inflation may be sensitive to the precise values of these price indices. However, this is only the case for persistent (low frequency) movements in these data series. Short-lived (high frequency) movements in these variables are largely interpreted as idiosyncratic movements. To demonstrate this we increase by 1 standard deviation the 2003Q1 values of these series. 19 Figure 5 shows the overall change in core CPI in 2003Q1 is approximately 0.1 percentage points or 0.15 standard deviations with smaller changes for periods before and after this date. This demonstrates that core inflation is relatively unresponsive to noisy movements in individual data series, even when one of these data series is an important determinant of core. In addition, we can use the data decomposition to identify how each of these data series con- 19 We picked a time period in the middle of the available sample because this results in visually more dramatic results. The size of the revisions is similar if the data changes are made at the end of the sample. Hairdressing, veterinary services and appliance repairs data was increased by 1.2, 2.1 and 3.3 percentage points respectively.

16 Reserve Bank of New Zealand Analytical Note Series 16 tributes to the revision of core CPI. This is illustrated in Figure It turns out that veterinary services price data the series with only the second largest overall contribution leads to the largest revision in core CPI of around 0.07 standard deviations. This is because veterinary services prices are slightly less volatile than hairdressing and appliance repairs prices so that a larger proportion of the price movements in veterinary services is treated as core inflation. 3.2 A small structural model Structural macroeconomic models can be useful tools for forecasting because they are able to provide users with an economic interpretation behind the forecasts they produce. The data decomposition can illuminate how these economic interpretations are deduced from observable data. The model considered in this section is a small open economy model similar to Justiniano and Preston (2010) with a tradable and domestic producing sector. Households supply labour to domestic firms and consume both domestic and tradable goods. Firms are subject to nominal rigidities which result in sluggish price adjustments to changes in economic conditions and a role for monetary policy to stabilize prices. Appendix B. The equations for this model are detailed in Because this is a structural model, the behaviour of economic agents such as households, firms and the central bank is described in some detail. These details allow the user to identify how movements in observed economic variables are determined by changes in the behaviour of agents in various sectors of the economy. For example, some structural models are able to take a position on whether given the structure of the model itself the values of economic variables such as the exchange rate are justified by economic conditions. Economic variables in modern structural models are assumed to be driven by a range of structural shocks. These shocks are unobserved and are modelled as stochastic in nature. When 20 An oddity made clear by the data decomposition is that core CPI a concept defined as summarizing co-movements between many variables is estimated by a method (Kalman smoothing) that interprets information from each data series independently from the information contained in other data series. The fact that the Kalman smoother does not adjust its estimates of the factors when the data co-moves could be interpreted as suggesting that the smoother is an inappropriate tool to use to estimate the factor. It turns out that this is not the case when the parameters in the model are estimated on the data set. The process of estimation involves selecting parameters to maximize the likelihood that the model can explain the observable data. This is equivalent to minimizing the random components in the model. Given the structure of the model, minimizing the random components largely involves minimizing the idiosyncratic movements in variables that is, setting the parameters so that the Kalman smoother estimates factors that maximize the co-movement between the various data series. Therefore, so long as the data being used when smoothing is the same as the data used in estimation, using the Kalman smoother to estimate the factor should be appropriate.

17 Reserve Bank of New Zealand Analytical Note Series 17 designing these models from the basic behaviours of economic actors, structural shocks are each given unique economic interpretations that can be used to describe the subsequent economic fluctuations these models generate. One example is the monetary policy shock term. The structural model considered here has an equation representing the central bank s monetary policy rule how the central bank is typically estimated to behave in response to the forecasted economic outlook. This particular structural model has a rule under which the central bank responds to its forecast of next quarter s CPI inflation as well as the current output gap and exchange rate. In the model, interest rates will sometimes deviate from the level proposed by the model s rule. In some cases, these deviations may occur deliberately and in other cases, these deviations may occur because the central bank s assessment of the level of inflationary pressure turns out to have been incorrect. The size of these deviations is assumed to diminish gradually over several quarters. Conceptualizing monetary policy in this manner can be useful to assess the effects of policy deviating occasionally from a systematic rule (either deliberately or otherwise). There are many other unobserved shocks in the model. A shock decomposition is a common tool used to ascertain the model implied explanation of which shocks (each with their own economic interpretation) have caused observed economic fluctuations. The shock decomposition provides information on how the various underlying economic shocks once identified contribute to the model s forecasts. However, the shock decomposition even when applied over history leaves out an important aspect of the process: how these various underlying economic shocks are themselves estimated. Before producing the model s forecasts the Kalman smoother needs to select a combination of the possible structural shocks available within the model to explain the observed economic fluctuations over the available sample. Because the shock decomposition takes the structural shocks as given, the data decomposition can complement the shock decomposition by illustrating how observable data contributed to the estimates of these structural shocks. In the remainder of this section, we look at a single quarter in history, in this case 2002Q1. Table 1 compares the 2002Q1 data with the structural model s forecasts for 2002Q1 constructed with 2001Q4 data for the variables used to produce the structural model s forecasts. 21 For example, GDP and consumption growth were lower than had been forecast whereas CPI, domestic inflation and exports were above the model s forecasts. When producing regular forecasts it is often a useful exercise to investigate how new releases of data will affect the model s forecasts. 21 The differences column in table 1 may not perfectly match the values calculated from other columns due to rounding.

18 Reserve Bank of New Zealand Analytical Note Series 18 Table 1: Data and the previous forecast for 2002Q1 Variable Data Forecast made in 2001Q4 Difference GDP growth Consumption growth Export growth CPI inflation Non-tradable inflation Wage inflation Interest rates Exchange rate change Terms of trade change The black lines in Figure 7 shows the effects of this new information on the subsequent forecasts produced by the model. A black line at zero indicates no revision to the forecast in response to new data whereas a positive value indicates that the forecast was, at that point in time, revised upwards. This figure shows that simply adding the new quarter s data led to the forecast of the 90-day interest rate to be revised downwards by approximately 0.5 percentage points (50 basis points) in 2002Q4. Similarly, the forecast for GDP growth is initially revised downwards but is predicted to return to the previous forecast. Figure 7: Revisions to forecasts from a structural model Interest Interest rate Rate GDP growth GDP Growth CPI inflation CPI Inflation The black lines in each panel represents the revision to the model s forecast for each variable while the points each labelled with an X the new data itself.

19 Reserve Bank of New Zealand Analytical Note Series 19 While not visible on the figure, the model s inflation target assumes that the central bank will adjust interest rates over time to ensure that, eventually, CPI inflation will return to target (leading to an eventual forecast revision of 0). Figure 8: Shock decomposition of forecast revisions 1 Interest rate GDP growth CPI inflation Interest Rate GDP Growth CPI Inflation Consumption shock - Permanent productivity shock - Import price shock - Exchange rate shock - Wage markup shock - Domestic markup Shock - Tradable markup shock - Export markup shock - Export demand shock - Monetary policy shock - Government spending shock This figure shows a shock decomposition of the forecast revisions caused by introducing a new quarter of data in 2002Q1. The black lines represent the overall forecast revisions and the coloured bars represent the contribution to these black lines from individual economic shocks. To provide an economic interpretation of these forecast revisions we can use a shock decomposition which is detailed in Figure 8. The black lines in this figure are the same forecast revisions shown in Figure 7. The bars represent contributions to these forecasts from the new economic shocks inferred by the introduction of 2002Q1 data. The model interprets the changes as most likely to have arisen from a combination of a positive export demand shock (an increase in demand for New Zealand exports at any given price) and a negative consumption shock (a desire on the part of households to, at any given interest rate, consume a larger portion of their income in the current period). These shocks have offsetting effects on GDP growth, and hence interest rates. The model also identifies several smaller shocks, including an exchange rate shock (indicating the exchange rate has risen by less than suggested by economic fundamentals) and a government spending shock. Shocks also appear to contribute to explaining periods before the new data occurs (even though the historical data themselves are not changed). This occurs because the new 2002Q1 data suggests a different interpretation of history in terms of this model s economic shocks. The data decomposition can complement the shock decomposition by illustrating which of the new observed data are important in generating the revisions to the estimated shocks. Figure 9 shows how observable data (in bars) contributed the model s recommended interest rate (black line in the left panel) as well as the difference between this recommended rate and actual interest rates the monetary policy shock (the black line in the right panel). Interest rate data includes both the changes in monetary policy that are consistent with the model s monetary

20 Reserve Bank of New Zealand Analytical Note Series 20 Figure 9: Data decomposition of recommended monetary policy and the estimated monetary policy shock 0.4 Recommended interest rate Recommended Interest Rates 0.1 Monetary policy shock Monetary Policy Shock GDP growth - Consumption growth - Wage inflation - Interest rates - Exchange rate - CPI inflation - Non-tradable inflation - Terms of trade - Export growth policy equation as well as monetary policy shocks. The interest rate forecast revision in figure 7 is made up of the combined effect of the two black lines in the panels in figure 9. All observable data are important in determining the model s recommended interest rate because the model recommends responding to its own forecasts of future inflation (as well as the output gap and exchange rate). Since future inflationary pressures are partly caused by current economic conditions, new observable data can also change the model s recommended interest rate. In addition, because it is assumed that the central bank takes considerable time to bring actual monetary policy back in line with the model s rule it also changes the forecast of the size of the monetary policy shock (deviations from this rule). Figure 9 suggests that the bulk of the revisions to the recommended interest rate occur from interest rate, exchange rate and terms of trade data while being offset by the impact of consumption growth data. Actual consumption growth data was much lower than had been forecasted, however, the model interprets this downwards revision as suggesting that consumption growth will be stronger in the future to return consumption to previous levels. This bounceback behaviour of consumption is forecast because the model explains a large portion of the negative consumption surprise with a temporary positive government spending shock (displacing the need for as much private consumption). 22 As private consumption returns to previous levels this will result in higher inflation. This results in consumption data contributing to upward 22 Govenment spending data are not observed by the model, yet the model structure details a government sector. The behaviour of this sector is inferred from the available observable data. The pink bars in figure 8 detail the overall effects of the government spending shock. These bars are influenced by a variety of factors, not just the fall in private consumption data.

21 Reserve Bank of New Zealand Analytical Note Series 21 revisions to the path of recommended interest rates. Curiously, higher terms of trade data contribute to lower recommended interest rates. While the surprise to terms of trade data does indicate higher export prices, export volumes data is also an observable. Since this decomposition isolates the effects of each individual data series, the shocks introduced by new terms of trade surprise must leave export volumes unchanged (changes in export volumes data are dealt with in another part of the decomposition). To leave export volumes unchanged, the terms of trade data introduces both export markup (supply) and demand shocks. The net effect of these two shocks is a reduction in resource and inflationary pressures in the medium term which lowers the model s recommended interest rate. 23 The monetary policy shock by contrast, seems largely driven by non-tradable inflation, exchange rate and interest rate data. The higher than forecast non-tradable inflation data seems to revise the estimates of shocks over history, revising upward the recommended interest rate over history. As interest rate data over the same period did not change, this causes downwards revisions to the historical estimates of monetary policy shocks (monetary policy over the historical period now looks looser, relative to the model s rule, than had been previously realized). These revisions to the historical estimates also affect the forecast of monetary policy shocks because the model assumes that deviations from the recommended rate take time to be corrected. The model appears to interpret new consumption growth and terms of trade data as suggesting identical revisions to the forecasts of recommended and expected monetary policy with little expected deviations from the recommended rule. Non-tradable inflation data raise the model s forecasts of inflationary pressure and raise recommended interest rates more than the model s forecasts of actual interest rates. Exchange rate data by contrast, lower the forecast of actual interest rates by more than its revisions to dis-inflationary pressures lowers recommended interest rates. Interest rate data both indicates a reduction in inflationary pressures as well as a deviation from the model s recommended rule. The data decomposition can also be used to identify how data affects other estimated economic shock terms in the model. To illustrate this Figure 10 shows the data decomposition of the the tradable markup shock and the exports demand shock. The tradable markup shock can be 23 Once accounting for export volumes data, the overall impact of the higher terms of trade and export volumes data suggests a demand curve shift. However, export volumes data does not reverse the revisions to recommended interest rates that were inferred from terms of trade data. In addition to the export demand shock, export volumes data has led the model to infer additional shocks that lead to a reduction in the costs of producing exports offsetting the inflationary pressures caused by higher export demand. These shocks are not the same as the tradable markup shock shown in figure 10

22 Reserve Bank of New Zealand Analytical Note Series 22 Figure 10: Data decomposition of the tradable markup shock and the exports demand shock 0.3 Tradable markup shock Tradable Markup Shock 3 Exports demand shock Export Demand Shock GDP growth - Consumption growth - Wage inflation - Interest rates - Exchange rate - CPI inflation - Non-tradable inflation - Terms of trade - Export growth thought of as representing changes in competitive pressures in the tradable sector (therefore resulting in changes to tradable firms desired markups). The exports demand shock is a modelbased measure of demand for New Zealand exports. In the model, the CPI is a weighted combination of tradable and non-tradable prices. Since tradable inflation is unobserved by the model, this variable is calculated by the Kalman smoother as the (weighted) difference between the observed CPI and non-tradable inflation data. 24 As table 1 shows, both the CPI and non-tradable inflation data were higher than expected, however the non-tradable inflation surprise was much larger in magnitude than the CPI inflation surprise indicating that tradable inflation fell during that quarter. One explanation for the lower tradable inflation is that the tradable sector has become more competitive. An alternative explanation is that costs fell and some of the falls in costs were passed on to consumers. Costs in the tradable sector within the model are assumed to be a weighted average of import prices (in New Zealand dollars) and domestic prices. This cost structure is the result of assuming that tradable firms are retailers selling imported goods who face some (domestic) transport costs to get their goods to market. The black line in the left panel of figure 10 shows that a reduction in the mark-ups achieved by tradable firms is estimated by the model to be one of the explanations of the fall in tradable inflation. The view that competitive pressures have increased (leading to lower pricing pressures in the tradable sector) seems to be based on the new non-tradable inflation data. New information on non-tradable inflation has two channels through which to adjust the model s estimate of the tradable markup shock. The first channel is that higher non-tradable inflation, while holding the 24 These two data series are the only two that affect the estimate of tradable inflation.