Autocorrelation Problem Set Refer To The Attached Stocks Dat

Autocorrelation Problem Setrefer To The Attached Stocks Data Data

Autocorrelation Problem Setrefer To The Attached Stocks Data Data

Autocorrelation Problem Set Refer to the attached “stocks” data. Data include NYSE index values, GDP measured in billions of dollars, and time from. First, estimate the following equation using Ordinary Least Squares (OLS):

Assessment of First-Order Autocorrelation in Stock Data

To evaluate the presence of first-order autocorrelation in the stock data series, we begin by visually inspecting the data through a scatter diagram. Specifically, we plot the residuals against their lagged values, which allows us to observe any systematic patterns or relationships indicating autocorrelation. If the points on the scatter plot display a discernible linear trend, this suggests potential autocorrelation issues.

Using the residuals obtained from the initial OLS estimation, a scatter plot where residuals are plotted on the y-axis and residuals lagged by one period are on the x-axis provides a visual diagnostic. The absence of any apparent pattern implies little evidence of autocorrelation, whereas a pattern such as clustering along a line indicates the presence of autocorrelation (Gujarati & Porter, 2009).

Statistical Test for First-Order Autocorrelation: Durbin-Watson Statistic

Beyond visual inspection, a formal statistical test is necessary to determine whether autocorrelation is statistically significant. The Durbin-Watson (D-W) test is employed for this purpose. First, the data should be set with the time variable as the panel identifier using the command tsset time. This prepares the data for time-series analysis within statistical software such as Stata.

Next, the D-W statistic can be estimated with the command estat dwatson. The D-W statistic ranges between 0 and 4, where a value around 2 suggests no autocorrelation, values less than 2 indicate positive autocorrelation, and values greater than 2 indicate negative autocorrelation (Durbin & Watson, 1950). For example, a D-W value of 1.2 suggests the presence of positive autocorrelation.

Correction of Autocorrelation Using the Newey-West Method

If autocorrelation is detected, it violates the classical assumption of uncorrelated errors, which can lead to inefficient estimates and biased standard errors. To remedy this, the Newey-West robust standard errors can be applied, allowing for autocorrelation and heteroskedasticity in the residuals.

The Newey-West procedure adjusts the covariance matrix of parameter estimates, providing consistent standard errors. In practical application, this correction can be implemented via statistical software commands (Newey & West, 1987). For instance, in Stata, after the initial OLS regression, one can run:

regress y x1 x2, vce(newey, lag(#))

where # is the number of lags deemed appropriate based on the data's autocorrelation structure. Common choices for lag length depend on sample size and the pattern of autocorrelation.

Transforming Data Based on Autocorrelation Diagnostics

Optionalizing, if the test from part (b) indicates significant autocorrelation, the data can be transformed to remove this autocorrelation. Using the estimated D value, a generalized difference transformation can be performed, following the equation:

Yt - d  Yt-1 = (Xt - d  Xt-1)β + εt

where d is derived from the D-test, often approximated as the D statistic itself. The transformed model should be re-estimated, and residuals checked to determine whether autocorrelation persists.

Specifically, after transforming the data, residual analysis similar to the initial steps (scatter plots and D-W tests) should be undertaken. If autocorrelation remains, further differencing or specialized correction methods may be necessary.

Conclusion

Addressing autocorrelation in stock data is crucial for valid inference and efficient estimation. Visual assessments via scatter plots, combined with the Durbin-Watson test, provide a foundation for detecting autocorrelation. Correction techniques such as Newey-West standard errors help mitigate the bias arising from autocorrelation, ensuring more reliable statistical inferences. Transformations based on diagnostic results further refine the model, addressing persistent autocorrelation issues. The integration of these methods enhances the robustness of time series analyses in financial and macroeconomic data (Stock & Watson, 2015).

References

• Durbin, J., & Watson, G. S. (1950). Testing for Serial Correlation in Least Squares Regression: I. Biometrika, 37(3/4), 409–428.
• Gujarati, D. N., & Porter, D. C. (2009). Basic Econometrics (5th ed.). McGraw-Hill.
• Newey, W. K., & West, K. D. (1987). A Simple, Positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica, 55(3), 703–708.
• Stock, J. H., & Watson, M. W. (2015). Introduction to Econometrics (3rd ed.). Pearson.
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