Dummy Variables. Susan Thomas IGIDR, Bombay. 24 November, 2008

Transcription

1 IGIDR, Bombay 24 November, 2008

2 The problem of structural change Model: Y i = β 0 + β 1 X 1i + ɛ i Structural change, type 1: change in parameters in time. Y i = α 1 + β 1 X i + e 1i for period 1 Y i = α 2 + β 2 X i + e 2i for period 2 Solution: for a given break point τ, σ 2 unrestricted = n1 e 2 1i + n2 e 2 2i vs.σ 2 restricted = N=n1+n2 ɛ 2 i Critical value: F(k, N - 2k) Other types of structural change Type 2: change in constant terms (dummy variables) Type 3: change in distribution of errors Type 4: change in sets of coefficients

3 Testing for change in parameters in the sample (r_i r_f) (r_i r_f) =alpha_1 + beta_1 (r_m r_f) (r_i r_f) =alpha_2 + beta_2 (r_m r_f) (r_m r_f) T_1 (r_m r_f) Time, t

4 Type 2: change in constant terms A dummy variable, D k is a binary variable which takes the value of 0 or 1 when the condition is false or true. Example: D k = 0 if a boy child is born, D k = 1 if a girl child is born. Dummies are useful in changing the structure of the model depending upon the value of some conditioning variable. The simplest is to change the intercept term of the regression model. Example: Y i = weight, X i = height Y i = α 1 + βx i + e 1i, i = female Y i = α 2 + βx i + e 2i, i = male

5 Type 2: change in constant terms Y = alpha_1 + beta X + e Y Y = alpha_0 + beta X + e alpha_1 alpha_0 D=0 D=1 X_i = D=1 D=0

6 Change in intercept: test of mean Data: Group 1 Y i = µ + ɛ i Group 2 Y i = (µ + δ) + ɛ i µ = µ G1, µ + δ = µ G2 or δ = µ G2 µ G1 The regression can be estimated as: Y i = µ + δd i + ɛ i where D i = 0 for Group 1, D i = 1 for Group 2. Alternative model: Y i = µ 1 G 1 + µ 2 G 2 + e i G 1 = 1, ifi = Group 1, otherwise G 1 = 0 G 2 = 1, ifi = Group 2, otherwise G 2 = 0 However, H 0 in model 2 cannot ask whether α 1, α 2 = 0. Advantage of the dummy variable model: H 0 : δ = µ G1 µ G2 is a well posed test of whether the mean of G 1, G 2 are different.

7 Change in intercept: test of mean Data frame for model 1 [y x] = Y = Y Y Y n1 1 0 Y n Y n Y N 1 1» In1 0 I n2 I n2 + ɛ OLS solution:» = ˆµˆδ» N n2 n 2 n 2 1» n1 ȳ 1 + n 2 ȳ 2 n 2 ȳ 2» = ȳ 1 ȳ 2 ȳ 1

8 HW Use the normal equations of the OLS optimisation to show that»» ȳ = 1 ˆµˆδ ȳ 2 ȳ 1 What is the standard error of ˆµ, ˆδ?

9 Change in intercept: test of mean Data frame for model 2 Y =» In1 0 0 I n2 + ɛ OLS solution:» ˆµG1 ˆµ G2 " σ ˆµG1 σ ˆµG2 # = =» n1 0 0 n 2» σɛ/ n1 σ ɛ/ n2 1» n1 ȳ 1 n 2 ȳ 2 =» ȳ1 ȳ 2

10 Model choices when dealing with dummy variables Model 1: Y i = µ + δd i + ɛ i Model 2: Y i = µ G1 D G1 + µ G2 D G2 + e i Incorrect model: Y i = α + µ G1 D G1 + µ G2 D G2 + e i Data matrix for the models: Model 1 Model 2 Incorrect model»»» In1 0 In1 0 In1 I n1 0 0 I n2 I n2 0 I n2 I n2 I n2 In the third data matrix, the sum of the second and third columns add up to the first. This means the inverse of (X X) 1 cannot be calculated. Which in turn means that three coefficents cannot be estimated. Problem of multicollinearity: with dummy variables, coefficients for a comprehensive set of dummies cannot be estimated simultaneously with an intercept. Model can either contain a comprehesive set of dummy variables or an intercept.

11 Modelling the index of industrial production, IIP

12 Seasonality in the IIP data tmp$iip

13 Features of IIP Data has monthly frequency from April 1990 to Sep 2008 Appears to have an annual trend linear? non-linear? Appears to have seasonality. Expected patterns at regular intervals. Model suggestions: A different level for different years: year trend term Captures a level of IIP for a given year. For example,trend is denoted as 1 for 1990, 2 for 1991, 3 for 1992, etc. A different level for different months: month dummies Captures a level of IIP for a given month in a year. Each month has a dummy. For example, Jan t is 1 for January in any month, and 0 otherwise. Model 1: IIP t = α 0 + α 1 Y t + β 1 Jan t + β 2 Feb t β 11 Nov t + ɛ t

14 Model 1 Regression results Estimate Std. Error t value Pr(> t ) (Intercept) year Residual SE = F-stat(1, 220) = 3322 prob value = 2.2e-16 R-squared = Adjusted R-squared:

15 Explained vs. Actual data iip

16 Behaviour of serial dependence in residuals ACF

17 Full model 1 Regression results Estimate Std. Error t value Pr(> t ) (Intercept) year jan feb mar may jun jul aug sep oct nov Residual SE = F-stat(11, 210) = prob value = 2.2e-16 R-squared = Adjusted R-squared:

18 Interpreting the model The omitted dummy is Dec. Therefore, the jan coefficient value of is the additional shift for January in addition to the value for December. In this model, the January effect is: = From the model, the IIP level for January 1991 is IIP jan 1991 = =

19 Explained vs. Actual data iip

20 Behaviour of serial dependence in residuals ACF

21 Model 2 The trend and seasonality is non-linear Model suggestions: Fit the model on log(iip) Model 2: log IIP t = α 0 + α 1 Y t + β 1 Jan t + β 2 Feb t β 11 Nov t + ɛ t

22 Simple model 2 Regression results: Estimate Std. Error t value Pr(> t ) (Intercept) year Residual SE = F-stat(1, 220) = 9077 prob value = 2.2e-16 R-squared = Adjusted R-squared:

23 Explained vs. Actual data liip

24 Full model 2 Regression results: Estimate Std. Error t value Pr(> t ) (Intercept) year jan feb mar may jun jul aug sep oct nov Residual SE = F-stat(11, 210) = 1042 prob value = 2.2e-16 R-squared = Adjusted R-squared:

25 Explained vs. Actual data liip

26 Residual data ACF

27 Results, inference There is still a lot of serial dependence in the residuals of the model. This means that there is yet a lot of variance about the IIP which is to be captured.

28 Model 3 The dummy variables capture a The linearity is better captured by log changes in IIP from the previous year. y t = log(iip t, y 1 ) log(iip t, y 0 ) This is a standard data transformation used in the econometric literature for seasonally adjusting macro-economic data. Model suggestions: There is a trend. There is seasonality. Model 2: log IIP t,y1 /IIP t,y0 = α 0 + α 1 Y t + β 1 Jan t + β 2 Feb t β 11 Nov t + ɛ t

29 IIP YoY growth series giip

30 IIP YoY growth series giip

31 Simple model 3 Regression results: Estimate Std. Error t value Pr(> t ) (Intercept) gyear Residual SE = F-stat(1, 208) = prob value = 1.45e-4 R-squared = Adjusted R-squared:

32 Explained vs. Actual data giip

33 Residual data ACF

34 Full model 3 Regression results: Estimate Std. Error t value Pr(> t ) (Intercept) year jan feb mar may jun jul aug sep oct nov Residual SE = F-stat(11, 198) = prob value = R-squared = Adjusted R-squared:

35 Explained vs. Actual data giip

36 Residual data ACF

37 Model 4: an autoregressive model for IIP Time series models use information from previous periods of own data to explain the next. Example, an autoregressive model for IIP would take the form: IIP t = α + β 1 IIP t 1 + ɛ t This is called the Autoregressive model (AR) of order 1 because it has only one previous period variable as the explanatory variable for IIP t. More generic forms of AR models are: IIP t = α + β 1 IIP t β k IIP t k + ɛ t This is an AR(k) model with IIP from k previous periods to explain IIP t.

38 Model 4: an autoregressive model for IIP Model for yoy-growth in IIP: giip t = giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t giip t 16 + ɛ t σ ɛ = σ giip =

39 Explained vs. Actual data giip

40 Dependence in residual data ACF

41 Dependence in variance of residual data ACF

42 Cross-plot of giip vs. residuals t$resid

43 Cross-plot of giip vs. residuals-squared (t$resid * t$resid)