Intervention Analysis and Transfer Function Models

Transcription

1 Chapter 7 Intervention Analysis and Transfer Function Models The idea in intervention analysis and transfer function models is to generalize the univariate methods studies previously to allow the time path of a dependent variable to be affected by the time path of an independent or exogenous variable. If it is known that there is no feedback, these can be very effective tools for forecasting and hypothesis testing. The major limitation of these models is that they rule out any feedback. If feedback exists a more appropriate tool are the vector autoregresssion (VAR) models that we will cover next chapter. 7.1 Intervention Analysis Consider the following example that follows from Enders (2010). Let {y t } represent the quarterly total skyjackings. In January 1973 the US started installing metal detectors in all airports. Suppose we want to compare the mean value of{y t } for all t < 1973Q1 with the mean value of{y t } for all t 1973Q1. As simple test using the averages is not correct because successive values of{y t } are serially correlated. That is, some of the effects prior to 1973Q1 may be carried over to the post-intervention regime. Consider the model in Enders, Sandler, and Cauley (1990), y t = a 0 + a 1 y t 1 + c 0 z t + ε t, a 1 <1 (7.1) where ε t is a white-noise term, and z t is a dummy variable equal to one beginning in 1973Q1, zero otherwise. 1. For t < 1973Q1 the value of z t = 0, the intercept term is a 0 and the long-run mean of the series is a 0 /(1 a 1 ). 2. For t 1973Q1 the value of z t = 1, the intercept term is a 0 + c 0 and the new long-run mean is(a 0 + c 0 )/(1 a 1 ). The impact effect of metal detectors is given by c 0. Its significance can be assessed using a standard t-test. The long-run effect of the intervention is given by c 0 /(1 83

2 84 7 Intervention Analysis and Transfer Function Models a 1 ), which is equal to the new long-run mean (a 0 + c 0 )/(1 a 1 ) minus the original mean a 0 /(1 a 1 ). The various transitional effects can be obtained from the impulse response function. Equation 7.1 can be expresses as such that (1 a 0 L)y t = a 0 + c 0 z t + ε t (7.2) y t = a 0 1 a 0 L + c 0z t 1 a 0 L + ε t 1 a 0 L = a 0 + c 0 1 a 0 i=0 a i 1 z t i+ i=0 a i 1 ε t i (7.3) Equation 7.4 is the impulse response function, where we can see the responses of the {y t } sequence to the intervention. For time t, the impact of z t on y t is given by c 0. For the remaining impulses we have that dy t /dz t i = dy t+i /dz t, and z t+i = z t = 1 for all i>0. Then, if we differentiate Equation 7.2 with respect to z t 1 and update by one period, dy t+1 dz t = c 0 + c 0 a 1 (7.4) Likewise, given that z t+1 = z t+2 = =1, the entire impulse response function is dy t+ j = c 0 [1+a 1 + a 2 dz 1+ +a j 1 ]. (7.5) t Taking the limits as j, we obtain the long-run impact given by c 0 /(1 a 1 ). This example of intervention can also follow a more general ARMA(p,q) process, y t = a 0 + A(L)y t 1 + c 0 z t + B(L)ε t (7.6) with A(L) and B(L) being polynomials of the lag operator L. There are several possible ways to model the intervention function. Four popular cases are: 1. Pure jump. z t is equal to one after period τ, zero otherwise. 2. Pulse function. z t is equal to one when t = τ, zero otherwise. Of course, the effect may last for many periods given the autoregressive nature of the{y t } series. 3. Gradually changing function. The intervention z t = 0 before time τ, then starting at τ increase monotonically from zero to one during a specific time interval. Then stays at one. 4. Prolonged impulse function. The intervention z t = 0 before time τ, then jumps to one at τ and decreases monotonically to zero during a specific time interval. Then stays at zero. Notice that the effect of the interventions change if {y t } has a unit root. A pulse intervention will have a permanent effect on the level of a unit root process. A pure jump will act as a drift term. Finally, the intervention may affect the variable of interest with a delay d. That is,

3 7.2 Transfer Function Models 85 y t = a 0 + A(L)y t 1 + c 0 z t d + B(L)ε t (7.7) Steps in Intervention Analysis The following example follows from Enders (2010). Let the linear from of the intervention model be y t = a 0 + A(L)y t 1 + c 0 z t + B(L)ε t. (7.8) Notice that the model assumes that the coefficients are invariant to the intervention. A simple test this assumption is to estimate the most appropriate ARIMA(p,d,q) before and after the intervention. If the two models are significantly different, then this can be interpreted as evidence of time dependence of the estimated coefficients. The typical steps in most intervention studies are: 1. Use the longest data span (i.e., before or after τ) to estimate the most appropriate ARIMA(p,d,q) model. Make sure that{y t } is stationary. One option is to use the unit root test with exogenous breaks. 2. Estimate the models using the entire sample, including the effect of the intervention. 3. Perform a diagnostic test of the estimated equation. Diagnostic test is important because we merged pre- and post-intervention periods. A well estimated model has the following characteristics: a. The estimated coefficients should be of high quality. That is, (1) all coefficients should be statistically significant, (2) we should have a parsimonious model, (3) the AR coefficients should imply that the {y t } sequence is convergent. b. The residuals should approximate a white-noise process. c. The tentative model (i.e., the intervention function model) should outperform plausible alternatives. This can be done using the AIC and the BIC to select alternative intervention function models. 7.2 Transfer Function Models A natural extension of the intervention model is to allow the {z t } sequence to be something different than a deterministic dummy variable. Consider the following generalization: y t = a 0 + A(L)y t 1 +C(L)z t + B(L)ε t, (7.9) where C(L) in an additional polynomial in the lag operator L. The polynomial C(L) is known as the transfer function, and shows how the exogenous variable z t affects

4 86 7 Intervention Analysis and Transfer Function Models the time path of the endogenous variable y t. The coefficients c i of the polynomial C(L) are the transfer function weights. The key restrictive assumption here is that z t is an exogenous process that evolves independently of the y t sequence. Innovations in{y t } are assumed to have no effects on the sequence {z t }, such that E(z t ε t s ) = 0 for all values of t and s. z t is called a leading indicator if c 0 = 0 in the polynomial C(L)=c 0 + c 1 L+c 2 L 2 +. This means that the contemporaneous z t does not affect y t, and that previous values of the observations in z t can be used to predict the future values of the{y t } sequence. 1 In its more general form, the{z t } process follows the form: D(L)z t = E(L)ε zt (7.10) where D(L) and E(L) are polynomials in the lag operator L and ε zt is white noise. Estimation of the model in Equations 7.9 and 7.10 involves various steps that can be summarized as follows: 1. Estimate the model for the z t sequence in Equation 7.10 using the most appropriate ARMA specification. The Box-Jenkins approach can be used here. 2. Obtain a filtered y t sequence by applying the filter D(L)/E(L) to the original sequence y t. 3. Define the cross correlation between y t and z t i as ρ yz (i) cov(y t,z t i ) (σ y σ z ) (7.11) The plot of ρ yz (i) for different values of i is known as the cross-correlogram. This is similar in nature to the ADF. The idea in this step is to use the crosscorrelogram to help identify the form of of A(L) and C(L). The process is not simple, but you will eventually have to estimate the model: [1 A(L)L]y t = C(L)z t + e t (7.12) where e t is the error term, which is not necessarily white noise. Keep in mind that in this step you are using the filtered sequence y t from step The{e t } sequence in the previous step is an approximation of B(L)ε. Hence, the ACF of these residuals can suggest the appropriate form of the B(L) function. 5. In the last step you simultaneously estimate A(L), B(L), and C(L). You should compare between different candidates from steps 3 and 4. This process of estimating Equation 7.9 involves judgment on the part of the researcher. As such, Stata and most statistical packages do not have an automated procedure to estimate transfer function models. To estimate these models in Stata you may want to follow the steps in the article by McDowell (2002). 2 1 Examples of transfer function models are the macroeconomic conditions{z t } that affect agricultural production{y t }. 2 McDowell, A., From the help desk: Transfer functions. The Stata Journal, 2, pp

5 7.3 Limits to Structural Multivariate Estimation Limits to Structural Multivariate Estimation Enders (2010) points out two important difficulties in fitting a multivariate equation such as a transfer function model. 1. The resulting model may be overparameterized. 2. The assumption of no feedback from the {y t } sequence to the {z t } sequence is strong. The solution to the second problem is to allow for feedback of reverse causality and estimate a nonstructural model following Sims (1980). This is known as the vector-autoregression approach.