1 Graphical method of detecting autocorrelation. 2 Run test to detect autocorrelation

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1 1 Graphical method of detecting autocorrelation Residual plot : A graph of the estimated residuals ˆɛ i against time t is plotted. If successive residuals tend to cluster on one side of the zero line of the other, it is graphical indication of the presence of autocorrelation. As the first step toward identifying the presence of autocorrelation, it is a good practice to plot e i against time i and look at the clustering effect. Run test to detect autocorrelation A run is defined to be a succession of one or more identical symbols which are followed and proceeded be a different or no symbol at all In the run test the hypothese are; Let H 0 : There are no autocorrelation problem in the model. H a : There is autocorrelation present in the model. n 1 = number of positive symbol n = number of negative symbol n = n 1 + n R = number of runs Under the null hypothesis that successive outcomes are independent and assuming that n 1 0 and n 0 the number of runs is distributed normally with MeanE(R) = Varianceσ R = n1n n 1 + n + 1 n1n(n1n n1 n) (n 1 + n )(n 1 + n 1) If n > 5 then the appropriate test statistics is z = R E(R) V ar(r) N(0, 1) If the hypothesis of randomness is sustainable we should expect R the number of runs obtained to lie between [E(R)±1.96σ R] with 95% confidence. Decision rule : Dont reject H 0 if the number of runs R falls between [E(R) 1.96σ R R E(R) σ R] and accept otherwise. 1

2 3 Durbin-Watson test The multiple rgeression model with AR(1) error is given by: y t = β 0 + β 1x t1 + β x t + + β k x tk + ɛ t ɛ t = ρɛ t 1 + u t, 1 < ρ < 1 Durbin-Watson is the most popular test detecting autocorrelation which is developed by J. Durbin and G. S. Watson. Using Durbin-Watson we can test the hypothesis H 0 : ρ = 0i.e. the errors are not correlated H a : ρ 0i.e. the errors are correlated The Durbin-Watson test statistic is defined as follows; n (ˆɛ ˆɛt 1) d = n ˆɛ = ˆɛ t ˆɛ tˆɛ t 1 n ˆɛ = 1 = [1 ρˆɛtˆɛ t 1 ] ˆɛtˆɛ t 1 ˆɛ t ˆɛ t 1 where ˆɛ t and ˆɛ t 1 differ only by one observation so they are approximately equal. Therefore, d = (1 ρ) 3.1 Range of Durbin-Watson test statistic We know that the range of ρ is between -1 and 1. Since the Durbin-Watson d statistic is a function of ρ, the range of the test statistic is also dependent on the range of ρ. When Thus, 0 d 4. ρ = 0 then d = ρ = 1 then d = 0 ρ = 1 then d = 4 3. Mechanism of Durbin-Watson test Estimate the model by OLS and compute the residuals ɛ t as y t ˆβ 0 ˆβ 1x t1 ˆβ k x tk Compute the Durbin-Watson test statistic; T (ɛt ɛt 1) d = T t=1 ɛ t

3 Note : We have seen that 0 d 4. The exact distribution of d depends on ρ as well as the observations on x s. Durbin and Watson had showed that the distribution of d is bounded by two limiting distributions that give the critical values for the limiting distributions of d, namely d L and d U. To test the null hypothesis H 0 : ρ = 0 against H a : ρ > 0, we at first have to find the critical values for the Durbin-Watson statistic: d L and d U. We reject H 0 if d L d U, we do not reject if d L d U. If d L < d < d U the test is inconclusive. To test for negative autocorrelation (that is for H a : ρ < 0), we use 4 d. this is done when d is greatter than. If 4 d < d L we conclude that there is significant negative autocorrelation. If 4 d > d U we conclude that there is no negative autocorrelation. The test is inconclusive of d L < 4 d < d U. 3.3 Remarks about Durbin-Watson test The inconclusiveness of the Durbin-Watson test arisen from the fact that there is no exact small-sample distribution for the Durbin-Watson statistic d. From the estimated residuals we can obtain an estimate of the firstorder autocorrelation coefficient as T ˆρ = ɛtɛt 1 T t=1 ɛ t This estimate is approcimately equal to the one obtained by regressing ɛ t against û t 1 without a constant term. It has been shown that d is approximately equal to (1 ˆρ). As we have seen earlier, the range of ρ is from -1 to +1, the range of d is obtained as 0 to 4. When ρ takes a middle value 0 then d takes the value. Thus DW statistic approximately means that there is no first order autocorrelation. A strong positive autocorrelation means ρ is close to +1. Similarly, values of d close to 4 indicate a strong negative correlation, i.e. ρ is close to The Lagrangian Multiplier Test The Lagrangian Multiplier test is useful in identifying autocorrelation not only of the first order but of higher orders as well. Let us Consider the following; y t = β 0 + β 1x t1 + β x t + + β k x tk + ρɛ t 1 + u t The test for ρ = 0 can be treated as the LM test for the addition of the variable ɛ t 1. 3

4 4.1 Steps Estimate the regression model by OLS and compute its estimated residuals ˆɛ t Regress ˆɛ t against a constant x t1,..., x tk and ɛ t 1 using T 1 observations through T. Then the LM statistic can be calculated by (T 1)Rɛ where Rɛ is the R-squares from the auxilliary regression. Reject the null hypothesis H 0 : ρ = 0 of zero autocorrelation in favor of the alternative that ρ 0 if (1 T )R ɛ > χ 1,(1 α), the value of χ 1 in the chi-square distribution with 1 df such that the areato the right of it is 1 α and α is the significance level. 4. Remarks If there was autocorrelation in the residuals, we wouls expect ɛ t to be related to ɛ t 1. This is the motivation behind the auxilliary regression in which ɛ t 1 is included with all the independent variables in the model. Unlike the DW test, the LM test does not have inconclusiveness. However, the one drawback of the LM test is that it is a large sample test. We need at least a sample size of 30 for the test to be meaningful. RCODE:: > l i b r a r y ( c a r ) > data ( H a r t n a g e l ) > H a r t n a g e l y e a r t f r p a r t i c d e g r e e s f c o n v i c t f t h e f t m c o n v i c t m t h e f t NA NA NA NA NA NA > d u r b i n W a t s o n T e s t ( lm ( f c o n v i c t t f r + p a r t i c + d e g r e e s + m convict, d a t a=h a r t n a g e l ) ) l a g A u t o c o r r e l a t i o n D W S t a t i s t i c p v a l u e A l t e r n a t i v e h y p o t h e s i s : r h o!= 0 Higher order autocorrelation : In general, the pth order autoregressive process of the residuals is given by; y t = β 0 + β 1 x t1 + β x t + + β k x tk + ɛ t ɛ t = ρ 1 ɛ t 1 + ρ ɛ t + ρ 3 ɛ t ρ p ɛ t p + u t 4

5 4.3 detection using Lagrangian Multiplier: The Lagrangian Multiplier test for higher order autocorrelation is obtained by combinig the above two equations into y t = β 0 + β 1 x t1 + β x t + + β k x tk + ρ 1 ɛ t 1 + ρ ɛ t + ρ 3 ɛ t ρ p ɛ t p + u t The null hypotheses is that each of the ρs is zero, i.e. ρ 1 = ρ = = ρ p = 0gainst the alternative that at least one of them is not zero. Steps involved in Lagrangian Multiplier : Estimate the regression model by OLS and obtain its estimated residuals ˆɛ t Regress ˆɛ t against all the independent variables x t1,..., x tk plus ɛ t 1, ɛ t,..., ɛ t p. The effective number of observations is T p because t p defines only for the period p + 1 to T. Compute (T p)r ɛ from the auxilliary regression run in Step. If this exceeds χ p,1 α, the value of the chi-square distribution with p df such that the area to the right is 1 α then reject H 0 : ρ 1 = ρ = ρ 3 = = ρ p = 0 in favor of H 1 : at least one of the ρ is significantly different from zero. 5