Introduction to Time Series Regression Time series data is extremely common and forecasting is perhaps the most employable sub discipline in economics. It is also a field that has seen a revolution in the last 20 years. Better yet, data is readily available making paper writing much easier. Furthermore, the procedures the profession wants to see have become more clearly defined – there is a template students can follow. Within the discipline imposed by the template there is still plenty of room for creativity. This handout loosely follows the content of chapter 12 of Stock and Watson. We begin by examining the simplest time series regression - a first order autoregression. This simply means we regress a variable on one of its lags. .16 It is generally a good idea to plot the data you will be studying. This plot of .12 inflation was generated in eviews from the commands: .08 Create q 46.1 2004.1 Cfetch pzunew Genr inf = .04 4*d(log(pzunew)) Plot inf .00 The graph shows gradually -.04 rising inflation until 1980 or so and falling thereafter. This corresponds to a well-known -.08 structural change in monetary 1950 1960 1970 1980 1990 2000 policy. On Oct 6, 1979 Paul Volcker announced the Fed INF would control the money supply and not interest rates in an effort to bring inflation under control. The first order autoregressive model of inflation is LS d(inf) c d(inf(-1)) Dependent Variable: D(INF) Method: Least Squares Date: 04/15/03 Time: 14:06 Sample(adjusted): 1947:4 2000:1 Included observations: 210 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C -0.000463 0.001629 -0.284402 0.7764 D(INF(-1)) -0.182134 0.067176 -2.711316 0.0073 R-squared 0.034136 Mean dependent var -0.000437 Adjusted R-squared 0.029492 S.D. dependent var 0.023961 S.E. of regression 0.023605 Akaike info criterion -4.645261 Sum squared resid 0.115894 Schwarz criterion -4.613384 Log likelihood 489.7524 F-statistic 7.351235 Durbin-Watson stat 2.151828 Prob(F-statistic) 0.007262 We differenced the data for reasons explained later. A fourth order autoregressive model is LS d(inf) c d(inf(-1)) d(inf(-2)) d(inf(-3)) d(inf(-4)) Dependent Variable: D(INF) Method: Least Squares Date: 04/15/03 Time: 14:10 Sample(adjusted): 1948:3 2000:1 Included observations: 207 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C -0.000363 0.001504 -0.241720 0.8092 D(INF(-1)) -0.202177 0.070248 -2.878045 0.0044 D(INF(-2)) -0.372561 0.070958 -5.250449 0.0000 D(INF(-3)) 0.124972 0.070586 1.770475 0.0782 D(INF(-4)) -0.044090 0.069287 -0.636335 0.5253 R-squared 0.200928 Mean dependent var -0.000219 Adjusted R-squared 0.185105 S.D. dependent var 0.023948 S.E. of regression 0.021618 Akaike info criterion -4.806693 Sum squared resid 0.094405 Schwarz criterion -4.726193 Log likelihood 502.4928 F-statistic 12.69832 Durbin-Watson stat 1.990404 Prob(F-statistic) 0.000000 Adding unemployment produces an autoregressive distributed lag model: LS d(inf) c d(inf(-1)) d(inf(-2)) d(inf(-3)) d(inf(-4)) lhur(-1) lhur(-2) lhur(-3) lhur(-4) Dependent Variable: D(INF) Method: Least Squares Date: 04/15/03 Time: 15:13 Sample(adjusted): 1950:1 2000:1 Included observations: 201 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 0.006672 0.005521 1.208511 0.2283 D(INF(-1)) -0.338561 0.071643 -4.725677 0.0000 D(INF(-2)) -0.431820 0.075299 -5.734727 0.0000 D(INF(-3)) 0.017905 0.072459 0.247104 0.8051 D(INF(-4)) -0.076628 0.068315 -1.121686 0.2634 LHUR(-1) -0.016334 0.004572 -3.572454 0.0004 LHUR(-2) 0.012626 0.009012 1.401065 0.1628 LHUR(-3) 0.004539 0.009021 0.503131 0.6154 LHUR(-4) -0.001980 0.004592 -0.431099 0.6669 R-squared 0.284873 Mean dependent var 0.000194 Adjusted R-squared 0.255076 S.D. dependent var 0.022060 S.E. of regression 0.019040 Akaike info criterion -5.040853 Sum squared resid 0.069601 Schwarz criterion -4.892944 Log likelihood 515.6057 F-statistic 9.560489 Durbin-Watson stat 1.965237 Prob(F-statistic) 0.000000 We can check whether adding the unemployment variables is informative by conducting the F- Test that all are jointly insignificant. To do this in eviews, while the equation above is displayed, click view, coefficient tests, redundant variables and enter the four lags of unemployment in the dialogue box. Redundant Variables: LHUR(-1) LHUR(-2) LHUR(-3) LHUR(-4) F-statistic 6.101250 Probability 0.000121 Log likelihood ratio 24.05091 Probability 0.000078 This kind of test is so common it has a name “Granger Causality” and is automated in Eviews. Cause(4) d(inf) lhur Pairwise Granger Causality Tests Date: 04/15/03 Time: 15:14 Sample: 1950:1 2004:1 Lags: 4 Null Hypothesis: Obs F-Statistic Probability LHUR does not Granger Cause D(INF) 201 6.10125 0.00012 D(INF) does not Granger Cause LHUR 0.52608 0.71668 Note that we reject that unemployment does not “Granger Cause” a change in inflation to exactly the same degree in both tests because the tests are the same. As the text points out, the test is badly named. Granger informative would be better. We have shown that unemployment helps predict changes in inflation even if the lags of inflation are available. The Eviews print out also performed a different test. It ran unemployment on four lags of unemployment and four lags of the change in inflation. The change in inflation is not informative in predicting changes in unemployment. Causation requires exogeneity and neither inflation or unemployment can be said to be exogenous. We have used four lags arbitrarily. To select lag length use the Schwartz criteria. For example, if we run the above regression with three lags Dependent Variable: D(INF) Method: Least Squares Date: 04/15/03 Time: 15:16 Sample(adjusted): 1950:1 2000:1 Included observations: 201 after adjusting endpoints Variable Coefficient Std. Error t-Statistic Prob. C 0.005331 0.005367 0.993322 0.3218 D(INF(-1)) -0.342109 0.071489 -4.785486 0.0000 D(INF(-2)) -0.405157 0.068535 -5.911658 0.0000 D(INF(-3)) 0.036063 0.068039 0.530037 0.5967 LHUR(-1) -0.016935 0.004499 -3.764030 0.0002 LHUR(-2) 0.015377 0.007987 1.925363 0.0556 LHUR(-3) 0.000638 0.004609 0.138386 0.8901 R-squared 0.279143 Mean dependent var 0.000194 Adjusted R-squared 0.256848 S.D. dependent var 0.022060 S.E. of regression 0.019017 Akaike info criterion -5.052772 Sum squared resid 0.070159 Schwarz criterion -4.937732 Log likelihood 514.8036 F-statistic 12.52066 Durbin-Watson stat 1.975165 Prob(F-statistic) 0.000000 The Schwarz criteria weighs improvement in SSR against additional variables. The criteria is defined so that new variables increase the value unless the fit improves and SSR declines. The criteria uses the log of SSR and our SSR is a fraction so the Schwartz criteria is negative. Still, we find the smallest value. Here, -4.93< -4.89 so the three lag model is preferred. We would continue making these comparisons. Note that my comparisons use exactly the same data and the same number of observations. This is not automatic. I reset the sample to start in 1950 so that enough prior data exists to allow 3 or 4 lags. (With two lags the Schwartz criteria is –4.988 and is better still. With one lag it is –4.765 so 2 lags is correct.) (We could have avoided negative info criteria by defining inflation as 400*d(log(pzunew)). This would give whole numbers rather than fractions for inflation rates. ) The text notes that the Akaike criteria is commonly used but has a theoretical flaw that leads to too many lags. Stationarity We began by noting that the series for inflation rose then fell and that the switch occurs at the same time Fed changes policy. Estimating a single equation through both historical periods is apt to be misleading because the relationships we are estimating are unlikely to be stable or stationary. There are two forms of non-stationarity that are important: structural breaks and unit roots. Structural breaks are intuitive and dealt with in straightforward ways. If you believe Y = a + bX changes at a particular date, define a dummy variable d, that becomes 1 at that date and is zero before. Now estimate: Y 0 1 X 3 d 4 dX Conduct the F test that 3 4 0. If you are uncertain, define many d’s and conduct many F tests and refer to a table of critical values in the text for the QLR statistic. As it happens, Eviews provides a rather different test for the purpose so we shall leave this for now. The second form of nonstationarity is more subtle but no less important. Unit roots are a fairly subtle point in econometrics. It is a point that the profession tried to ignore as inconvenient for a time but one that we all now recognize as fundamental. Students often go through the same process, why torture us with one more problem? Isn’t checking for multicollinearity, heteroschedasticity and autocorrelation enough? After all it should be given the classical regression model. Unfortunately, much, perhaps most, economic data does not follow the classical regression model. Economic data is time series, the prior value powerfully affects the future value. Worse, much data is the value of an asset. If the asset’s value is expected to rise, people will buy it and the value rises until further increases are not predicted. Therefore rational expectations and efficient markets require that asset values be unpredictable. If the change in an asset’s value is unpredictable then the changes are random and may be written as (1) pt pt 1 where is a normally distributed random error. Rearranging we have (2) pt pt 1 which allows us to recognize the series as a random walk. Random walks are quite intuitive. Consider a person at a particular place. If they take a step, they are where they were plus a one step movement in some direction. That is precisely what the equation says. If the person takes enough steps, even if the steps themselves are random, they may end up at a place arbitrarily far from the initial point. Now consider two random walkers both beginning from Denver. Each flips a fair coin and if the toss is a head, takes one step north, if tails, one step south. We stop them 1 million steps later. One eventually finds himself in Canada and the other swimming in the Gulf of Mexico. If we record their position as distance from the equator, a regression would uncover a negative correlation that is completely spurious. If they had both ended up in Canada the regression would claim a positive correlation for a purely random and unpredictable event. Indeed only if one of the walkers miraculously finds themself back in Denver will the regression give the correct result that the two walks are uncorrelated. If all series are random walks, the solution is easy, regress the change in position on the change in position. Knowing that on the 50,000th trial walker A took a step north will tell you nothing about walker B. There is a closely related time series process with very different properties. Consider (3) pt a bpt 1 , b 1. If we want to continue with the random walker analogy the story becomes rather strange. Think of a hot air balloon with an elastic band connecting it to the equator and engines that push it north. The parameter b measures the weakness of the elastic band. For b = 1 the band is so weak the balloon stays wherever it was the previous period. For b = .9, the band is strong enough to pull the balloon 10% of the way to the equator. The parameter a measures the strength of the engines pushing north. At some point, the strength of the engines is just enough to offset the pull from the elastic band. At this long run equilibrium point, pt pt 1 and pt (1 b) a or a pt . 1 b 1 b Clearly, such process is in deep trouble if b 1. Our balloon is pushed farther and farther north by its engines while the elastic band has no ability to pull it back. Eventually, it will find itself well beyond the North Pole and in deep space while the error term becomes arbitrarily large as well. These facts are commonly referred to by the mystic muttering “unit roots produce non- stationary processes that are not mean reverting and have variances that increase with the number of observations.” Mean reversion just means the process has an equilibrium it returns to although shocks may push it away temporarily. The process above is mean reverting only if b 1. It is easy to give (3) an economic interpretation. Consider the following model: YCI C a bY(-1) Substituting we find C a bI b(C(-1)) We hope the system converges to an equilibrium where C = C(-1), if so C - b(C(-1)) a bI a bI C . 1 b 1 b To summarize: Random walks are formally rather similar to convergent processes like the multiplier. Regressing random walks on each other is very likely to produce spurious regression results. Therefore before conducting regressions it is important to at least know whether the series are stationary. This process will take a number of forms that we will explore soon, but certainly commonsense plays a role. Is the series an asset value that theoretically ought to be a random walk? Is the model underlying the process a convergent series? Before moving on to testing for unit roots, it is a good idea to play around with regressions where we know the structure to get some feeling for the seriousness of the problem. The following program is easily run in Eviews. Create u 1000 smpl 1 1 genr a = 10 genr b = 5 genr x = 10 genr y = 5 smpl 2 1000 genr a = a(-1) +nrnd genr b = b(-1) +nrnd genr x = .80*x(-1) +nrnd genr y = .80*y(-1)+nrnd smpl 1 1000 graph g1.line a b graph g2.line x y equation eq1.ls a c b equation eq2.ls x c y You should be able to see the mean reversion in x and y but not a and b. The regression of b on a will likely generate a strong t-stat indicating an association that does not exist while the t-stat for y will likely be small. Try it. 12 60 40 8 20 4 0 0 -20 -4 -40 -60 -8 250 500 750 1000 250 500 750 1000 A B X Y Dependent Variable: A Method: Least Squares Date: 04/15/03 Time: 16:24 Sample: 1 1000 Included observations: 1000 Variable Coefficient Std. Error t-Statistic Prob. C 13.86998 0.396701 34.96335 0.0000 B -0.121270 0.016461 -7.367050 0.0000 Dependent Variable: X Method: Least Squares Date: 04/15/03 Time: 16:24 Sample: 1 1000 Included observations: 1000 Variable Coefficient Std. Error t-Statistic Prob. C -0.314450 0.054595 -5.759645 0.0000 Y -0.007032 0.030604 -0.229767 0.8183 Unit Root Testing So, we need a test to help distinguish which of the two we have, non-mean reverting, non- stationary unit roots or convergent series. What follows is informal. I strongly recommend reading the sections in the Eviews manual on unit roots. (Starts on pg. 325 of users guide.) Recall that the problem is that yt ayt 1 t and we want to know if a is too close to 1 or not. The accepted procedure, known as the Dickey Fuller Test, is to run d ( yt ) yt 1 t and test if 0. If 0 , then a = 1. If instead, we can reject the null, then the series is not unit root. There are many problems. First, the t-stat for this test, given the hypothesis of a unit root does not have the normal critical values. Eviews will display the correct critical values for each test it runs. Next, we need the residual to be white noise. Autocorrelation is treated by adding lagged difference terms. So the test equation might be: d ( yt ) yt 1 i d ( yt 1 ) t with enough lags to remove autocorrelation. This is referred to as the Augmented Dickey-Fuller Test. It may also be necessary to include an intercept or time trend. Mei Chu Hsiao recommends always adding an intercept and putting in the time trend if it is statistically significant. She suggests using the Schwartz or Akaike criteria to select number of lags. Stock and Watson suggest using the Akaike criteria because it is known to provide too many lags and for this test, the problems from too many lags are not severe. They also point out that linear trends are not the only alternative to stochastic trends. It may be that a non-linear trend is the appropriate alternative hypothesis in which case they refer you to a more advanced text. In the exercise you will see how to get Eviews to print out a summary of several tests for lags. The Eviews manual is less clear. They prefer a theory based decision but are aware this may not be possible.
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