ECON 604 EVIEWS TUTORIAL #2 IDENTIFYING AND ESTIMATING ARMA MODELS Note: We will use EViews 5.1 in this tutorial. There may be minor differences in EViews 6. Preliminary Analysis In this tutorial we will identify and estimate an ARMA model for the quarterly change in business inventories, stated at annual rated in billions of dollars. We have 60 observations covering the period 1955.1 to 1969.4. The data is available at the course web page. First obtain the data file business_inventory.wf1 and open it. Select the series inventory in the work file, then select [Quick/Graph/Line graph]: Type inventory in the box if it is not already written: 1
Click OK, you should see the following graph 20 16 12 8 4 0-4 -8 1956 1958 1960 1962 1964 1966 1968 Change in Business Inventories A causal examination of the graph suggests that the series is stationary (no apparent trend, and no change in volatility). The observations seem to fluctuate around a fixed mean, and the variance seems to be constant over time. Identification Close the graph, and open the inventory series by double clicking on the inventory in the workfile. Click [View], select [Correlogram] and type 15 in the Lags to include box, and click OK. Box and Jenkinks (1976) suggest that most autocorrelations we may safely examine is about one fourth of the number of observations. With 60 observations we may calculate 60/4 = 15 2
autocorrelations. Now, you should see the ACF, PACF, the portmanteau Q statistics, and its p value in the following window. Consider the estimated ACF in the above figure. Only the first 6 autocorrelations are significantly different from zero at 5% level: only the first five autocorrelations extend beyond the 95% confidence intervals indicated by dashed lines. The autocorrelations decay to statistical insignificance rather quickly. We conclude that the mean of the series is probably stationary. An AR model seems appropriate because the ACF decays toward zero rather than cutting off sharply to zero. If the ACF cuts off to zero, it suggests a moving average model. A decaying ACF is also consistent with a mixed ARMA model. But starting with a mixed model is often unwise for three reasons. First, it is often difficult to correctly identify a mixed model initially. Mixed nature of the model is more likely to be detected at the diagnostic stage. Second, the principle of parsimony suggests that we first try a simple AR or MA model before considering a mixed model. Finally, starting immediately with a mixed model may result in parameter redundancy (or common factors problem). We attentively propose an AR model. Experience and the shape of the PACF, which cuts off to zero after lag 1, suggest that we should consider p = 1 or p = 2 as an initial choice. The PACF cuts off to zero after lag one, so it is more likely that we have an AR(1) model. Thus, we suggest to estimate the following model: or 3
where. Estimation We first estimate an AR(1) model as suggested by the ACF and PACF. We will estimate the intercept c and φ 1 simultaneously note that / 1, so we can recover μ from c and vice versa. In order estimate the AR(1) model, select [Quick/ Estimate Equation] and type inventory c ar(1), and click OK. In EViews ar(k) stands for the kth lag of the modeled series, here inventory. That is ar(k)=inventory t k. You should get the following estimation output: Dependent Variable: INVENTORY Method: Least Squares Date: 10/18/07 Time: 21:32 Sample (adjusted): 1955 Q2 1969Q4 Included observations: 59 afte r adjustments Convergence achieved after 3 iteratio ns Variable Coefficient Std. Error t-statistic Prob. C 6.191556 1.418264 4.365588 0.0001 AR(1) 0.689752 0.095688 7.208318 0.0000 R-square d 0.476872 Mean dependent var 6. 123729 Adjusted R-squared 0.467694 S.D. dependent var 4.631390 S.E. of regression 3.379030 Akaike info criterion 5.306365 Sum squared resid 650.8172 Schwarz criterion 5.376790 Log likelihood -154.5378 F-statistic 51.95985 Durbin-Watson stat 2.095747 Prob(F-statistic) 0.000000 Inverted AR Roots.69 Click [Name] and type eq_ar1 for the name to save this model as an EViews equation for later use. In this output, it is very important to note that EViews reports rather than as an estimate of the intercept. Here the estimated AR(1) parameter is 0.689752, while the estimated intercept is given by 1 1 0.689752 6.191556 1.920918. 4
The model certainly satisfy the stationary requirement 1. The estimated AR(1) parameter is also significantly different from zero since for the hypothesis : 0, the t statistic is given by 0 se 0.689753 0.095688 7.208318 and much larger than the critical value 2.0. The Schwarz information criterion for this model is 5.3768. In order to compare with ARMA(p,q) models with, 2, we estimate each combination an report the corresponding Schwarz information criterion in the following table. We note that in order estimate a moving average term the EViews keyword ma(k) should be use, where k is the lag. For instance ma(1)=ε t. In order estimate these models, we proceed as in the preceding case by replacing the equation to estimate as follows: AR(2): MA(1): MA(2): ARMA(1,1): ARMA(2,1): ARMA(1,2): ARMA(2,2): inventory c ar(1) ar(2) inventory c ma(1) inventory c ma(1) ma(2) inventory c ar(1) ma(1) inventory c ar(1) ar(2) ma(1) inventory c ar(1) ma(1) ma(2) inventory c ar(1) ar(2) ma(1) ma(2) For instance in order to estimate the ARMA(2,1), select [Quick/Estimate Equation] and type the equation as follows: This should give the following estimation output: 5
Dependent Variable: INVENTORY Method: Least Squares Date: 10/18/07 Time: 22:34 Sample (adjusted): 1955Q3 1969Q4 Included observations: 58 after adjustments Convergence achieved after 12 iterations Backcast: 1955Q1 1955Q2 Variable Coefficient Std. Error t-statistic Prob. C 6.302821 1.896556 3.323298 0.0016 AR(1) 0.153273 0.491479 0.311860 0.7564 AR(2) 0.557357 0.353897 1.574911 0.1212 MA(1) 0.474353 0.492646 0.962868 0.3400 MA(2) -0.271391 0.193717-1.400967 0.1671 R-squared 0.495501 Mean dependent var 6.129310 Adjusted R-squared 0.457426 S.D. dependent var 4.671639 S.E. of regression 3.441112 Akaike info criterion 5.391729 Sum squared resid 627.5862 Schwarz criterion 5.569353 Log likelihood -151.3601 F-statistic 13.01370 Durbin-Watson stat 1.970862 Prob(F-statistic) 0.000000 Inverted AR Roots.83 -.67 Inverted MA Roots.34 -.81 We obtain the following Schwarz information criteria (SC) for these models AR(1): 5.376790 AR(2): 5.453999 MA(1): 5.557594 MA(2): 5.560201 ARMA(1,1): 5.432318 ARMA(2,1): 5.450277 ARMA(1,2): 5.493857 ARMA(2,2): 5.391729 The minimum SC value is obtained for the AR(1) model. Thus we consider this model as the ideal candidate. Our next step is to check it s appropriateness. 6
Diagnostic Checking To determine if the AR(1) model is statistically adequate, we first compare its theoretical ACF and PACF to empirical ACF and PACF of the data. Now open eq_ar1 and click [View] and choose [ARMA Structure/Correlogram], type 15 for the Lags and click OK: We obtain the following graphics:.8 Autocorrelation.6.4.2.0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Actual Theoretical Partial autocorrelation.8.6.4.2.0 -.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Actual Theoretical 7
Clearly, the theoretical ACF and PACF matches all the significant sample ACF and PACF estimates. The AR(1) model produces quite similar autocorrelation function with a merging of sampling error. A second diagnostic check we will do is the independence of the random shock terms εt. In order to check this, we use residual autocorrelation function. Click [View], chose [Residual Tests/Correlogram Q statistics], and type 15 in the box for Lags to include. We obtain the following picture: The Ljung Box Q statistic for the testing : 0 is 8.5291 with a p value of 0.860. The independence of random shocks is certainly not rejected even at 15 lags. We also note that all residual autocorrelation estimates are within the 95% confidence intervals. We safely assume that random shock term is white noise under an AR(1) model. Therefore, AR(1) model is appropriate for the quarterly changes in business inventories. 8