Download Methods to Remove Autocorrelation - Econometric Modeling - Lecture Notes and more Study notes Econometrics and Mathematical Economics in PDF only on Docsity!
ui = ρ ui‐ 1 + ε t For the runs test, in Column VI of the above table, the total number of positive values (N 1 ), and the total number of negative values (N 2 ) are counted. We also count that R=80 times ui changed signs from positive to negative and then from negative to positive. So, for the Runs test, we have N 1 = 677, N 2 = 720, N=N 1 +N 2 = 1397, and R = 80. So, E(R) = + 1 = + 1 = 698. Var(R) = = = 348. Then, for 95% confidence level (or equivalently, 5% significance level), if observed R falls between E(R)±1.96σR , then there is no autocorrelation, but if it falls outside, it can be said that autocorrelation exists. Now, E(R) + 1.96σR = 698.84 + 1.96 * √348.34 = 735. and E(R) - 1.96σR = 698.84 ‐ 1.96 * √348.34 = 662. But observed R is 80, and it lies outside the interval [662.26, 735.42] So, again we infer that there IS autocorrelation. So now, we can say that we cannot believe the significances of the βi s that we got from OLS. We first have to remove the autocorrelation and then find the significance to make correct conclusions about which factors affect gold prices.
That is, our detection of autocorrelation is over, and we can now move to the next step, that is, removal of autocorrelation. There are 4 methods to remove autocorrelation:
- First difference Method (applicable here, since d<R^2 )
- Find ρ from Durbin Watson d statistic, and use Generalised Least Squares Regression
- Find ρ from Eqn. 2, and use Generalised Least Squares Regression
- Use iterative Methods like Cochrane‐Orcutt (need to use softwares for this)
We see that the Durbin Watson d statistic shows that there is no autocorrelation in the first difference equation. Now we can believe the values of βi s. t statistic for β 2 is less than 2, so it is statistically insignificant => USD exchange rate does not affect gold price significantly; t statistic for β 3 is more than 2, so it is statistically significant => Sensex affects gold price significantly and negatively; and t statistic for β 4 is more than 2, so it is statistically significant => Oil price affects gold price significantly and positively. Method 2: Find ρ from Durbin Watson d statistic, and use Generalised Least Squares Regression From the original OLS, we got Durbin Watson d‐statistic as: 0.039. It has been established that for large samples, d ≈ 2(1‐ρ). Then, ρ ≈ 1 ‐d/2 = 1 ‐0.039/2 = 0. Gold price (^) t = β 1 + β 2 * USD exchange ratet + β 3 * Sensext + β 4 * oil pricet (‐) ρ * Gold price (^) t‐ 1 = ρ * β 1 + ρ * β 2 * USD exchange rate (^) t‐ 1 + ρ * β 3 * Sensex (^) t‐ 1 + ρ * β 4 * oil price (^) t‐ 1
dcalc
X
(Gold pricet ‐ ρ * Gold pricet‐ 1 ) = (β 1 ‐ ρ * β 1 ) + β 2 * (USD rate (^) t ‐ ρ * USD rate (^) t‐ 1 ) + β 3 * (Sensext ‐ ρ * Sensext‐ 1 ) + β 4 * (oil pricet ‐ ρ * oil pricet ) t =^ β 1 *^ (1‐^ ρ)^ +^ β 2 *^ t +^ β 3 *^ t +^ β 4 *^ t where (^) t = (Gold pricet ‐ ρ * Gold pricet‐ 1 ), (^) t = (USD ratet ‐ ρ * USD rate (^) t‐ 1 ), (^) t = (Sensex (^) t ‐ ρ * Sensex (^) t‐ 1 ) and (^) t = (oil pricet ‐ ρ * oil pricet) Now we compute the above values in the following table: The results from the OLS are as follows: = 25.293 * + (‐0.182) * + 0.165 * Std Error 19.1136 0.027703 0. t 1.323284 ‐6.56367 6. R^2 0. Durbin Watson 1.775142 dL = 1.645 (From Durbin‐ dU = 1.692 Watson Tables)
That is ρ = 0.979 ≈ 0.98, same as Case 2. The results of the Generalized Least Squares also are the same. Method 4: Iterative Method, specifically the Cochrane‐Orcutt procedure The algebra here is more involved, so I shall not write about it. In fact, this method is normally available in any software that handles Time Series Analysis. Here I give the output : Gold price =
25.801 * USD Exch. Rate
(‐0.181 ) * Sensex + 0.165 * oil price Std Error 3267.98069 19.119 0.028 0. t 1.116 1.349 ‐6.554 6. R^2 0. Durbin Watson 2.04 dL = 1.645 (From Durbin‐ dU = 1.692 Watson Tables) Here also Durbin Watson d statistic shows that there is no autocorrelation in the generalized equation. Now we can believe the values of βi s. t statistic for β 2 is less than 2, so it is statistically insignificant => USD exchange rate does not affect gold price significantly; t statistic for β 3 and β 4 are more than 2, so they are statistically significant => Sensex and oil price affect gold price significantly. Final comment: From the OLS of Eqn 1, all coefficients were highly significant. But autocorrelation was detected and removed, after which only 2 coefficients remained significant. I hope this clarifies why it is necessary to detect and remove autocorrelation.
REFERENCES FOR FURTHER READING:
Dielman, Terry E.: Applied Regression Analysis for Business and Economics, PWS‐Kent, Boston, 1991. Draper, N. R., and H. Smith: Applied Regression Analysis, 3d ed., John Wiley & Sons, New York, 1998. Frank, C. R., Jr.: Statistics and Econometrics, Holt, Rinehart and Winston, New York, 1971. Goldberger, Arthur S.: Introductory Econometrics, Harvard University Press, 1998. Graybill, F. A.: An Introduction to Linear Statistical Models , vol. 1, McGraw‐ Hill, New York, 1961. Greene, William H.: Econometric Analysis, 4th ed., Prentice Hall, Englewood Cliffs, N. J., 2000. Griffiths, William E., R. Carter Hill and George G. Judge: Learning and Practicing Econometrics, John Wiley & Sons, New York, 1993. Gujarati, Damodar N.: Essentials of Econometrics, 2d ed., McGraw‐Hill, New York, 1999. Hill, Carter, William Griffiths, and George Judge: Undergraduate Econometrics, John Wiley & Sons, New York, 2001. Hu, Teh‐Wei: Econometrics: An Introductory Analysis, University Park Press, Baltimore, 1973. Johnston, J.: Econometric Methods, 3d ed., McGraw‐Hill, New York, 1984. Katz, David A.: Econometric Theory and Applications, Prentice Hall, Englewood Cliffs, N.J., 1982. Klein, Lawrence R.: An Introduction to Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1962. Koop, Gary: Analysis of Economic Data, John Wiley & Sons, New York, 2000. Koutsoyiannis, A.: Theory of Econometrics, Harper & Row, New York, 1973. Maddala, G. S.: Introduction to Econometrics, John Wiley & Sons, 3d ed., New York, 2001. Mills, T. C.: The Econometric Modelling of Financial Time Series, Cambridge University Press, 1993. Mittelhammer, Ron C., George G. Judge, and Douglas J. Miller: Econometric Foundations, Cambridge University Press, New York, 2000.
d) Plot the error terms against time
- This is not an important assumption for computing the d statistics a) The regression model includes an intercept term b) The explanatory variables are fixed in repeated sampling c) The error terms are generated by the first order auto regressive scheme d) The error terms are not correlated with each other
SELF EVALUATION TESTS/ QUIZZES
1. In a study of the determination of prices of final output at factor cost in the United
Kingdom, the following results were obtained on the basis of annual data for the period
PFt = 2.033 + 0.273Wt − 0.521Xt + 0.256M t + 0.028M t-1 + 0.121PFt-
se = (0.992) (0.127) (0.099) (0.024) (0.039) (0.119)
R
2
= 0.984 d = 2.
Where PF = prices of final output at factor cost, W = wages and salaries per employee, X
= gross domestic product per person employed, M = import prices, M t− 1 = import prices
lagged 1 year, and PF t− 1 = prices of final output at factor cost in the previous year.
“Since for 18 observations and 5 explanatory variables, the 5 percent lower and upper d
values are 0.71 and 2.06, the estimated d value of 2.54 indicates that there is no positive
autocorrelation.’’ Comment
2. Given a sample of 50 observations and 4 explanatory variables, what can you say about
autocorrelation if (a) d = 1.05? (b) d = 1.40? (c) d = 2.50? (d) d = 3.97?
3. Give circumstances under which each of the following methods of estimating the first-
order coefficient of autocorrelation ρ may be appropriate:
a) First-difference regression
b) Moving average regression
c) Theil–Nagar transform
d) Cochrane and Orcutt iterative procedure
e) Hildreth–Lu scanning procedure
f) Durbin two-step procedure
4. State whether the following statements are true or false. Briefly justify your answer.
a) When autocorrelation is present, OLS estimators are biased as well as inefficient.
b) The Durbin–Watson d test assumes that the variance of the error term ut is
homoscedastic.
c) The first-difference transformation to eliminate autocorrelation assumes that the
coefficient of autocorrelation ρ is −1.
d) The R^2 values of two models, one involving regression in the first difference form
and another in the level form, are not directly comparable.
e) A significant Durbin–Watson d does not necessarily mean there is autocorrelation of
the first order.
