I'm confused. Why do you have regression residuals? I guess there are two times in this project where you would apply a white noise process. First, looking at a stationary series is that stationary series white noise? You would apply this directly to the sample autocorrelations of the series. The second time you would use it would be after you fit the ARIMA model. If the model is a "good fit" then the residuls between the actual and the modeled data should form a white noise process.
As far as the three tests I'm not completely sure what to tell you. Could you possibly be performing the DW test wrong or on the wrong residuals? I think they mention in the book that you could have some that say white noise and some not which would make you use the third test... or something like that. I guess it doesn't give you complete confidence that it isn't white noise, but you have a good idea since two of the three tests show it to be so.
Jacob: Suppose we take first differences to convert a random walk to white noise. Can we use the Durbin-Watson statistic or the Box-Pierce Q statistic to test for white noise, or must we first fit an ARIMA model?
Rachel: Examine the residuals using the Durbin-Watson statistic or the Box-Pierce Q statistic. If you have not fit an ARIMA model, the residuals are the first differences minus the mean.
Jacob: Is the mean zero?
Rachel: If the random walk has a drift of zero, the mean of the first differences is zero. Otherwise, the mean of the first differences is the drift of the random walk.
Jacob: How do we form the Durbin-Watson statistic in Excel? Is there is built-in function?
Rachel: Excel has no built-in function for this, but the formula is simple; see below.
Jacob: The formula uses the residuals. How do we calculate the residuals? We can do this from the equations in the textbook, but it would take a while. Is there a simple method?
Rachel: The Excel regression add-in calculates the residuals. The add-in computes the ordinary least squares estimators and the residuals. You can copy the formula for the Durbin-Watson statistic from the illustrative spreadsheet on the NEAS web site.
Jacob: Can you give an example?
Rachel: We form an AR(1) model from the last 3½ years of Treasury bill interest rates: January 1997 through June 2000. The correlogram indicates that this time series is a random walk, which is not stationary. Your student project might use first differences, not the interest rates themselves. But we start with the interest rates and compute the Durbin-Watson statistic to see if any serial correlation is significant. We also compute the slope coefficient and use the t statistic to test if it is significantly different from one.
Jacob: How do we use the regression add-in?
Rachel: Copy the January 1997 through June 2000 Treasury bill rates to a new worksheet. Place these in cells B11:B52 and also in cells C12:C53. Column B is the Y values and Column C is the X values. We don’t use the values in B11 or C53.
On the illustrative worksheet, we eliminate rows 53 and 11, getting rid of the original cell B11 and cell C53. This gives a matrix of B11:C51 for the regression analysis.
If we use an AR(2) model, we also place these rates in cells D13 : D54. Column B is the Y values, Column C is the X1 values, and Column D is the X2 values. We don’t use the values in B11, B12, C12, C53, D53, or D54.
Jacob: What do we choose on the regression menu for the AR(1) model?
Rachel: The dependent variable is in cells B11:B51, after eliminating the original cell B11. The independent variable is in cells C11:C51. You can place the output on the same sheet or a new sheet. Ask for residuals. You don’t need the standardized residuals since the interest rates are all about the same value in this scenario. If interest rates change greatly over the time series, we would examine standardized residuals.
The residual output shows the observation number, the fitted Y value, and the residual. In a new worksheet, these are in Columns A, B, and C. We placed the output on the same worksheet starting in cell A61.
In column D, place the square of the residual. For cell D85, write =C85^2. Copy this formula to cells D86 : D125.
In column E, place the difference of successive residuals. For cell E86, write =D86-D85. Copy this formula to cells E87:E125.
In Column F, place the square of the difference of successive residuals. For cell F86, write =E86^2. Copy this formula to cells F87:F125.
Use Excel’s quick sum function to add the numbers in Columns D and F. Place the cursor in cell D126 and click on the quick sum icon. Do the same for cell F125. The Durbin-Watson statistic is the sum in cell F125 divided by the sum in cell D126. We place the Durbin-Watson statistic in cell G126.
Jacob: What do we expect to find?
Rachel: A Durbin-Watson statistic of 2 indicates no serial correlation. This example gives a Durbin-Watson statistic of 2.10, which is not significantly different from 2. The autocorrelation of lag 1 on the residuals is –0.06289 (cell C127), which is not significantly different from zero.
Other time periods show significant serial correlation. Your student project should explain the meaning of your results.