Autocorrelation S. K. Bhaumik Reference: Sankar Kumar Bhaumik, Principles of Econometrics: A Modern Approach Using Eviews, Oxford University Press, 2015, Ch. 5 1 ❖ Definition • One of the assumptions of the CLRM → the disturbance term of the model is independent. Symbolically, for the model Yt = + X t + t Cov( t , s ) = E( t s ) = 0 for t s This feature of regression disturbance is known as serial independence or non- autocorrelation. It implies that the value of disturbance term in one period is not correlated with its value in another period → the disturbance occurring at one point of time does not carry over into another period. 2 • However, this assumption may not always be fulfilled, particularly when we are dealing with time series data. This is because the disturbance captures the impact of a large number of random and independent factors, which enter into our relationship but are not measurable. Therefore, it is quite possible that the effect of these factors operating in one period would carry over, at least in part, to the following period → the disturbance term at period t may be related with the disturbance terms at t − 1, t − 2, ... and t + 1, t + 2, ...., and so on. • In that case, Cov( t , s ) 0 for t s and we say that the disturbances are auto-correlated. • So autocorrelation represents lack of independence for the disturbance term of the model as a result of which its value in one period gets correlated with its value in another period. 3 ❖ Sources of autocorrelation ➢ Prolonged influence of shocks: In time series data, random shocks or disturbances have effects which persist for a number of time periods → an earthquake, flood, drought, or war affects the performance of the economy in periods following the period of its occurrence. The influences of such shocks become all the more visible when we consider shorter time-intervals (in weekly, monthly and quarterly data compared to yearly data) → this problem becomes more prevalent when time series data with shorter time intervals are considered. ➢ Inertia: Past actions often have strong influence on current actions owing to inertia or psychological conditioning → a positive disturbance in one period is likely to influence activities in successive periods resulting in autocorrelation. 4 ➢ Spatial autocorrelation: Although autocorrelation is found more in time series data, some cross-sectional data might also suffer from this problem. For example, while working with household level data, we may observe the disturbances of the households in the same neighbourhood being correlated. This is called ‘spatial autocorrelation’. ➢ Model misspecification: Exclusion of a relevant explanatory variable from the model, whose successive values are correlated, will make the disturbance term associated with the misspecified model autocorrelated. So when that variable is included in the model, the apparent problem of autocorrelation disappears. 5 ❖ Specification of Autocorrelation Relationship Suppose two successive values of disturbance term are correlated so that the autocorrelation relationship is t = t−1 + ut (1) where is a parameter whose absolute value is less than 1, i.e., 1 and ut is the disturbance term of the above autocorrelation relationship, which has the following properties. E(u ) = 0 t for all t 2 2 Var(ut ) = E(ut ) = u ut is normally distributed E(ut ut−i ) = 0 Equation (1) is known as the first-order autoregressive scheme, denoted by AR(1) . 6 By successive substitutions for t−1, t−2 , .... in (1), we obtain t = ( t−2 + ut−1 ) + ut 2 = t−2 + ut−1 + ut 2 = ( t−3 + ut−2 ) + ut−1 + ut 3 2 = t−3 + ut−2 + ut−1 + ut = + + 2 + .... ut ut−1 ut−2 (2) 1 This shows that under the first-order autoregressive scheme, the effect of past disturbances wears off gradually as . We may use the expression (2) to compute mean, variance and covariance of disturbances. Mean of t The mean of is 2 E( ) = E(u + u −1 + u −2 + .... ) t t t t = 0 = 0 ( E(ut ) for all t) 7 Variance of 2 2 Var( t ) = E(ut + ut−1 + ut−2 + .... ) 2 2 2 4 2 2 = E(ut + ut −1 + ut −2 + .... + 2ut ut −1 + 2 ut ut −2 + ....) 2 4 = Var(ut ) + Var(ut−1 ) + Var(ut−2 ) + .... ( E(ut ut−i ) = 0) 2 2 4 2 2 = u (1+ + + .... ) ( E(ut ) = u for all t) 2 u = 1− 2 2 = (3) Covariance of and t−1 From (2), we have t 2 t = ut + ut−1 + ut−2 + .... Lagging this by one period, = u + u + 2u + .... t−1 t−1 t−2 t−3 8 So the covariance between and is Cov( t , t−1 ) = E( t t−1 ) 2 2 = E[{ut + ut−1 + ut−2 + ....}{ut−1 + ut−2 + ut−3 + ....}] 2 = E[{ut + (ut−1 + ut−2 + ....}{ut−1 + ut−2 + ut−3 + ....}] 2 2 2 = E[ut ut−1 + ut ut−2 + ut ut−3 + ....] + E[ut−1 + ut−2 ut−3 + ....] 2 2 2 4 2 = ( u + u + u + .... ) (using assumptions of ut mentioned above) 2 = u 1− 2 2 2 2 2 = (σε = σu (1 − ρ ),as shown in (3) above) (4) From (4), it follows that E(εt εt−1) ρ = 2 (5) σε t−1 2 E(εtεt−2 ) ρ = 2 σε t . s E(εtεt−s ) ρ = 2 (6) σε 9 An important observation → if we put s = 0 in (6), then 2 2 2 2 E(εt ) σε = σε σε = 1 which means that zero-order autocorrelation coefficient is unity. We can rewrite (5) as Cov( , ) = t t−1 Var( t ) Var( t−1 ) This means that also represents the correlation coefficient between and t−1 . 2 s Similarly, is t the correlation coefficient between and t−2 and is the correlation coefficient between and t−s . 10 ❖ Consequences of Autocorrelation ✓ Unbiasedness Consider the model ˆ for which , the OLSY estimator= + of X, is+ given by t x y t t ˆ = t t 2 xt xt t = + 2 xt Taking expectations, E(ˆ) = ˆ is an unbiased estimator of . So autocorrelation does not remove unbiasedness property for the OLS estimators. 11 ✓ Bestness To examine this property we compute the variance of under autocorrelation, which is compared with variance of when there is no autocorrelation. ˆ ˆ 2 Var() = E[ − ] 2 x = E t t 2 xt 1 = ˆ E x 2 2 + 2 x x + 2 x x + .... 2 2 t t t t−1 t t−1 t t−2 t t−2 ( xt ) t=1 t=2 t=3 1 2 2 2 2 2 = 2 2 xt + 2 xt xt−1 + 2 xt xt−2 + .... ( xt ) E( t t−k ) k k 2 = E( t t−k ) = 2 2 xt xt−1 2 xt xt−2 = 2 1+ 2 2 + 2 2 + .... (7) xt xt xt Since is positive and xt xt−k is also positive (expectedly), the expression in the bracket is greater than 1. 12 Var(ˆ) Var(ˆ) ˆ Thus, Under AR No AR is no longer a minimum variance or best estimator. ✓ Consistency limVar(ˆ) = 0 Here we have to check whether n→ . From (7), we have 2 ˆ xt xt−1 2 xt xt−2 Var() = 2 1+ 2 2 + 2 2 + .... xt xt xt 2 2 2 2 = 2 + 2 2 xt xt−1 + xt xt−2 + .... xt ( xt ) 2 2 2 2 2 2 2 2 = 2 + 2 2 r1 xt xt−1 + r2 xt xt−2 + .... xt ( xt ) xt xt−k 2 2 r = x x − = r x x − k 2 2 t t k k t t k xt xt−k 2 2 / n 2( / n) 2 2 2 2 2 = 2 + 2 2 r1 xt / n xt−1 / n + r2 xt / n xt−2 / n + .... xt / n ( xt / n) 13 2 Now as n → , / n → 0 2 2 2 xt / n, xt−1 / n, xt−2 / n, .... approach to some finite number (say ), and * * r1 , r2 , .... approach to numbers with absolute value 1, say r1 , r2 , .... Therefore, lim( 2 / n) 2 lim( 2 / n) ˆ n→ n→ * 2 * lim Var( ) = + 2 (r1 + r2 + ....) = 0 n→ ˆ is a consistent estimator even under autocorrelation. 14 Tests for Autocorrelation ❖ Durbin-Watson (1951) test → simplest and most widely used Assumptions: ▪ The regression tmodel= t −includes1 + ut an intercept term. ▪ We are examining presence of first-order autocorrelation. ▪ The regression model doesn’t include a lagged dependent variable as an explanatory variable. Test procedure: Suppose the model is Y = + X + X + ... + X + t 0 1 1t 2 2t k kt t (8) ρ 1 (9) [ ut has all usual OLS assumptions] The null hypothesis being tested is: H N : ρ = 0 is called first-order autocorrelation coefficient. 15 Steps Step 1: Estimate model (8) by OLS and obtain the estimated residuals ˆt = et . Step 2: Compute Durbin-Watson d-statistic as n ˆ ˆ 2 ( t − t−1 ) t=2 d = n ˆ 2 t t=1 n 2 (et − et−1 ) t=2 = n 2 et t=1 n 2 n 2 n et + et−1 − 2 et et−1 = t=2 t=2 t=2 n 2 et t=1 For large samples, n 2 n n 2 et − 2 et et−1 et et−1 d t=1 t=1 21 − t=1 n 2 n 2 et et t=1 t=1 n 1 et et−1 Or d 2 (1 − ρ) = n t=1 (10) 1 n 2 1 n 2 et et−1 n t=1 n t=1 16 The value of d* lies between 0 and 4 Testing statistical significance of d* ▪ Compare d* with the values of theoretical-d (upper and lower), available from Durbin-Watson d-table.