1 Quantitative Techniques in Practice

Transcription

1 1 Quantitative Techniques in Practice 1.1 Lecture 2: Stationarity, spurious regression, etc Overview In the rst part we shall look at some issues in time series economics. In the second part we shall look at an application: estimation of the capital asset pricing model (CAPM) The time series issues are: 1. stationarity and non-stationarity: 2. spurious correlation 3. diagnostic tests 4. estimation: rst di erences and co-integration Stationarity and non-stationarity Many data series have a tendency to change over time: world population, national income, the price level, share prices. The way in which they change will vary in detail - but they show no tendency to stick around a particular range of values. Such variables are said to be nonstationary. Econometricians and people doing advanced stu in nance need to de ne di erence degrees of stationarity, but for our purposes we can stick with the de nition in Ashenfelter, Levine, and Zimmerman (pp ). The time series of observations Y 1 ; Y 2 ; Y 3 ; ::: etc. can be thought of being the outcome drawings from a joint probability density function f(y 1 ; Y 2 ; Y 3 ; :::Y T ). Stationarity implies that this function does not change as we move through time i.e. f(y 1 ; Y 2 ; Y 3 ; :::Y T ) = f(y 201 ; Y 202 ; Y 203 ; :::Y 200+T ) (1) Note that this does not require the Y 0 s to be independent of each other: Y t can be a function of its lagged values, in the tradition of so-called autoregressive models. If we think the assumption of exact equality in the f 0 s is too strong we can invoke weak stationarity: 1

2 1. E(Y t ) = constant mean of Y t : 2. Constant variance of Y t 3. E[(Y t ; )(Y t+k )] = k - the covariance between two values of Y separated in time by k periods is independent of time. For example, consider the autoregressive model Y t = k + 1 Y t Y t Y t 3 + u t (2) The s are the lag coe cients. Whether condition 1 is satis ed depends primarily on the sum of the lag coe cients. This is illustrated in the Excel le Lec2Stationarity.xls. (To be sure, the value of k also matters. For this exercise we have adjusted it to give a constant value in the absence of the shock u t :) The critical parameter is the sum of the lag coe cients. If this is less than 1 we have a stationary series. If 1:00 the series is non-stationary. Table 2.1 Case According to this Cases 2 and 3 should be nonstationary; Cases 1 and 4 stationary. You can verify this by playing with the spreadsheet. An example is reproduced below in gure 2.1 Figure 2.1 Spot the two nonstationary series. Spurious correlation What happens if we regress one nonstationary series on another? I tried it with two realisations of Case 3 above. Here is the output: SUMMARY OUTPUT 2

3 Stationary and nonstationary series Y(t) Period Series1 Series2 Series3 Series4 3

4 Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 26 Coefficients Standard Error t Stat Intercept X Variable Even though the two realisations of case 3 were based on independent statistical processes we appear to have a signi cant econometric relationship (e.g. the t value of 70 is much bigger than the critical value of for 24 degrees of freedom.) By contrast, here is the output from two adjacent realisations of model 4: Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 26 Coefficients Standard Error t Stat Intercept X Variable The lesson is: regressing one non-stationary series on another can lead to spurious regression results. What can be done? Well, if we take rst di erences i.e. de ne a new variable = Y t Y t 1 in a non-stationary series it may become stationary. Such a non-stationary series is said to be integrated of order 1or I(1) for short. Most non-stationary series are of this type. But sometimes we may have to take the di erence of that di erence to get a stationary series. This would then be integrated of order 2 or I(2) for short. 4

5 The "unit root" concept Related to the idea that = 1 represents an critical region is the idea of a "unit root." This represents a process known as a random walk. (see pp of Ashenfelter et.al.) Statistical tests, in particular the Dickey-Fuller (DF) and Augmented Dickey Fuller (ADF) tests, have been derived to check whether a data series has a unit root. ("Augmented" refers to testing given the possibility of lags.) For an example see below. Diagnostics and tests No single test will tell you all you need to know about a data series. should do a number of things: So you 1. Graph the series. What does it look like? 2. Ask EViews for the correllogram. Click on the data series. Then ask for View...Correllogram.. This gives raw and partial correlations of the series with its lagged values. How regular does the correlogram look? [Cue OHP transparencies showing correllograms for cases 1,2,3,] 3. Carry out a unit root test,asking for View...Unit root test... Use the Augmented Dickey-Fuller test. Here is an output for Case 1 above (key numbers in bold): Null Hypothesis: CASE1 has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=8) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Ignore this bit of output for the time being: Augmented Dickey-Fuller Test Equation Dependent Variable: D(CASE1) Method: Least Squares 5

6 Date: 01/28/04 Time: 14:31 Sample(adjusted): 1 25 Included observations: 25 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. CASE1(-1) C R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Because < we reject the null hypothesis of a "unit root" By contrast, here is the output for a realisation of Case 2: Null Hypothesis: CASE2 has a unit root Exogenous: Constant Lag Length: 8 (Automatic based on SIC, MAXLAG=8) t-statistic Prob.* Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Warning: Probabilities and critical values calculated for 20 observations and may not be accurate for a sample size of 17 Because > (closer to zero) we accept the null hypothesis. Now look what happens when we try Case3: Null Hypothesis: CASE3CORR has a unit root Exogenous: Constant Lag Length: 0 (Automatic based on SIC, MAXLAG=8) t-statistic Prob.* Augmented Dickey-Fuller test statistic

7 Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. Like all tests the power of the test (ability to correctly reject a null hypothesis) depends on the true situation. For Case 3 which in a sense is "more" in the nonstationary region than case 2, the acceptance of the unit root is much clearer. Estimation One answer to spurious correlation to to make sure we use stationary series. e.g. by di erencing our non-stationary variables we can produce stationary variables that will not be subject to spurious correlation. Alternatively, we can make use of the fact a linear combination of nonstationary series will tend to be non-stationary. An example of the such a combination is, from the regression model: Y t = a + bx t + u t (3) the disturbance term u t = Y t a bx t (4) will tend to be non-stationary if the regression is spurious. Hence the residual e t will be nonstationary. However, if X and Y are both nonstationary because they are related to each other causally according to the model it may be that u t and hence e t are stationary. The combination of a and b is known as a cointegrating vector. One way to test for a co-integrating relationship is to conduct a DF or ADF test on the residuals. (However, be warned that the critical values are di erenct than for the test we have just used, because we are using an estimate of the true disturbance term.) Reading Chapter 17 of Ashen eter, Levine, and Zimmerman. Salvatore & Reagle, chapter 11 7