Introduction Recursive residuals Structural change CUSUM and CUSUMSQ. Recursive residuals by Tabar and Andrea

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1 Recursive residuals by Tabar and Andrea

2 Least-squares estimator In the context of Least-Squares Estimation (i.e. OLS residuals), residuals can be find with this equation e = Y Xb. They are mainly used as a fitting criterion to make the sample regression line as close as possible to the data points by minimizing the squared-sum of the residuals, that is min e e = (Y Xb) (Y Xb). b

3 Recursive residuals Also recursive residuals are defined as the difference between Y and Xb, but calculated without using all the observations. We can calculate recursive residuals by using the first t 1 observations. The t th recursive residual is defined as the difference between Y t and X t b t 1.

4 Recursive residuals Also recursive residuals are defined as the difference between Y and Xb, but calculated without using all the observations. We can calculate recursive residuals by using the first t 1 observations. The t th recursive residual is defined as the difference between Y t and X t b t 1. The t th standardized recursive residual is obtained by dividing the t th recursive residual by its forecast variance.

5 Pros Unlike OLS residuals, standardized recursive residuals have different interesting proprieties. They are homoskedastic

6 Pros Unlike OLS residuals, standardized recursive residuals have different interesting proprieties. They are homoskedastic They are indipendent of one another

7 Pros Unlike OLS residuals, standardized recursive residuals have different interesting proprieties. They are homoskedastic They are indipendent of one another

8 Pros Unlike OLS residuals, standardized recursive residuals have different interesting proprieties. They are homoskedastic They are indipendent of one another Both properties make standardized recursive residuals attractive to calculate some regression diagnostics.

9 Cons A note of caution is presented by Kennedy:... because the behavior of recursive residuals in a mis-specified model is very different from that of the OLS residuals, (... ) test procedures based on the recursive residuals should be viewed as complementary to tests based on OLS residuals.

10 Recursive residuals Suppose that the sample contains a total of T observations.

11 Recursive residuals Suppose that the sample contains a total of T observations. The t th recursive residual is the ex-post prediction error for y t when the regression is estimated using only the first t 1 observations e t = y t x t b t 1. (1)

12 Recursive residuals Suppose that the sample contains a total of T observations. The t th recursive residual is the ex-post prediction error for y t when the regression is estimated using only the first t 1 observations The forecast variance of this residual is e t = y t x t b t 1. (1) σ 2 ft = σ 2 [1 + x t (X t 1X t 1 ) 1 x t ]. (2)

13 Recursive residuals The t th stanadardized recursive residual is w t = e t x t (X t 1 X t 1) 1 x t (3)

14 Structural change Recursive residuals can be used both to test for non-linearity and to test for structural change.

15 Structural change Recursive residuals can be used both to test for non-linearity and to test for structural change. Kennedy provides a simple explanation of the use of recursive residuals to test for non-linearity based on the concept of the U-shaped that suggest that there is a structural change.

16 CUSUM and CUSUMSQ To test a structural stability of the model there are different tests based on recursive residuals. The two most important are the CUSUM and the CUSUM-OF-SQUARES, with the data ordered chronologically, rather than according to the value of an explanatory variable.

17 CUSUM and CUSUMSQ To test a structural stability of the model there are different tests based on recursive residuals. The two most important are the CUSUM and the CUSUM-OF-SQUARES, with the data ordered chronologically, rather than according to the value of an explanatory variable. The CUSUM test id based on a plot of the sum of the recursive residuals. If this sum goes outide a critical bound, one concludes that there was a structural break at the point at which the sum began its movement toward the bound.

18 CUSUM and CUSUMSQ To test a structural stability of the model there are different tests based on recursive residuals. The two most important are the CUSUM and the CUSUM-OF-SQUARES, with the data ordered chronologically, rather than according to the value of an explanatory variable. The CUSUM test id based on a plot of the sum of the recursive residuals. If this sum goes outide a critical bound, one concludes that there was a structural break at the point at which the sum began its movement toward the bound. The CUSUM-OF-SQUARES test is similar to the cusum test, but plots the cumulative sum of squared recursive residuals, expressed as a fraction of these squared residuals summed over all observations.

19 CUSUM The CUSUM test is based on the cumulated sum of the residuals: with W t = T j=k+1 w t ˆσ T j=k+1 (w t w) 2 ˆσ 2 = T k 1 and w = T j=k+1 w t T k, where k is the minimum sample size for which we can fit the model.

20 CUSUM The CUSUM test is performed by plotting W t against t. Under the null hypotesis, W t, the cumulative sum, with constant parameter has a mean of zero, E(W t ) = 0, and variance equal to the number of residuals being summed, in fact each term has variance 1, and they are indipendent. But with nonconstant parameter, W t will tend to diverge from the zero mean value. The significance of the departure from the zero line may be assessed by reference to a pair of straight lines that pass through the points and (k, ±a t k) (t, ±3a t k), where a is a parameter depending on the significance level α chosen.

21 CUSUMSQ The second test statistic, the CUSUMSQ, is based on cumulative sums of squared residuals: with S t = t k+1 w 2 j T k+1 w 2 j t = k + 1,, T The expected value of S t is E(S t ) = t k, which goes to zero at T k t = k. The significance of departures from the expected value line is assessed by reference to a pair of lines drown parallel to the E(S t ) line at a distance c s above and below. This value depends on both the sample size T k and the significance level α.

ECON 5350 Class Notes Functional Form and Structural Change

ECON 5350 Class Notes Functional Form and Structural Change 1 Introduction Although OLS is considered a linear estimator, it does not mean that the relationship between Y and X needs to be linear. In this