BEE2006: Statistics and Econometrics

Transcription

1 BEE2006: Statistics and Econometrics Tutorial 2: Time Series - Regression Analysis and Further Issues (Part 1) February 1, 2013 Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

2 10.1 (a) Like cross-sectional observations, we can assume that most time series observations are independently distributed. Do you Agree or Disagree? Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

3 Consider the following two models Return i = β 0 + β 1 GDP i + u i Return t = β 0 + β 1 GDP t + u t Return i is the stock market returns at time t of country i Return t is the stock market returns of country i at time t GDP i is the GDP at time t of country i GDP t is the GDP of country i at time t

4 Would it be natural to expect: Corr (u i, u s GDP) =0 Corr (u t, u s GDP) =0 i s t s Suppose that if the stock market drastically decreased in period t 1(thinkaboutsomeoilshocku t 1 ), the government afraid of recession actively intervenes and shocks the stock market with some stimulus u t. u t = α 0 + α 1 u t 1 + e t then we ll have autocorrelation.

5 Would it be natural to expect: u i N ( 0,σ 2) u t N ( 0,σ 2) A lot of research in time series is devoted to the idea of Autoregressive conditional heteroskedasticity σ 2 t = α 0 + α 1 e 2 t α q e 2 t q + δ 1 σ 2 t δ p σ 2 t p

6 Example of clustering:

7 10.1(b) The OLS estimator in a time series regression is unbiased under the first three Gauss-Markov assumptions. Do you Agree or Disagree? Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

8 The first three assumptions: y t = β 0 + β 1 x 1t β k x kt + u t Assumption 1: Linear in Parameters Assumption 2: E (u t X) =0 t =0, 1, 2,...,n E (u t x 1t,...,x kt )=E (u x t )=0 Assumption 3: No perfect Collinearity Corr (x jt, x it ) 1 j i and t =1, 2, 3,...,n THEN THE OLS IS UNBIASED

9 10.1(c) A trending variable cannot be used as the dependent variable in multiple regression analysis. Do you Agree or Disagree? Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

10 Suppose your model y t = β 0 + β 1 x t + u t looks like this

11 There is obviously at time trend (upward) you should have consider this model: y t = β 0 + β 1 x t + β 2 t + u t Then β 2 captures the changes in y t caused by x t isolating for the time trend

12 10.1(d) Seasonality is not an issue when using annual time series observations. With annual data, each time period represents a year and is not associated with any seasons. Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

13 10.2 Let ggdp t denote the annual percentage change in gross domestic product and let int t denote a short-term interest rate. ggdp t = α 0 + δ 0 int t + δ 1 int t 1 + u t Assume that: E (u t int t, int t 1, int t 2,...,int 0 )=0 Cov (u t, int t )=0fort, t 1, t 2, t 3,...,0 Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

14 Suppose that the Federal Reserve seeks to control interest rate by the rule int t = γ 0 + γ 1 (ggdp t 1 3) + v t γ 1 > 0 Corr (v t, u t )=0forallt Corr (v t, int t )=0forallt show that Cov (u t 1, int t ) 0 and as a consequence E (u t int) 0 since E (u t 1 int) 0

15 From we can get ggdp t = α 0 + δ 0 int t + δ 1 int t 1 + u t ggdp t 1 = α 0 + δ 0 int t 1 + δ 1 int t 2 + u t 1 then int t = γ 0 + γ 1 (α 0 + δ 0 int t 1 + δ 1 int t 2 + u t 1 3) + v t Rearranging we have that int t =(γ 0 + γ 1 α 0 3γ 1 )+γ 1 δ 0 int t 1 +γ 1 δ 1 int t 2 +γ 1 u t 1 +v t

16 Now find Cov (u t 1, int t )= Cov (u t 1, (γ 0 + γ 1 α 0 3γ 1 )+γ 1 δ 0 int t 1 + γ 1 δ 1 int t 2 + γ 1 u t 1 + v t ) Recall that: Cov (u t 1, int t 1 )=0 Cov (u t 1, int t 2 )=0 Cov (u t t, v t )=0

17 Cov (u t 1, int t )=Cov (u t 1,γ 1 u t 1 )=γ 1 V (u t 1 ) Assume that V (u t 1 )=σ 2 homoskedasticity Then Cov (u t 1, int t )=γ 1 σ 2 0 since γ 1 > 0

18 10.6(a) Consider the following General Model: y t = α 0 + δ 0 z t + δ 1 z t 1 + δ 2 z t 2 + δ 3 z t 3 + δ 4 z t 4 + u t Now assume that we have a specific polynomial distribution lag δ j = γ 0 + γ 1 j + γ 2 j 2 where j are the quadratic lag. Eg. δ 2 = γ 0 + γ 1 2+γ Plug δ j into the model and rewrite the model in terms of parameter γ h for h =0, 1, 2 Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

19 We Know that: δ 0 = γ 0 δ 1 = γ 0 + γ 1 + γ 2 δ 2 = γ 0 +2γ 1 +4γ 2 δ 3 = γ 0 +3γ 1 +9γ 2 δ 4 = γ 0 +4γ 1 +16γ 2

20 Rewrite the model we get y t = α 0 + γ 0 (x 1t )+γ 1 (x 2t )+γ 2 (x 3t )+u t where x 1t = z t + z t 1 + z t 2 + z t 3 + z t 4 x 2t = z t 1 +2z t 2 +3z t 3 +4z t 4 x 3t = z t 1 +4z t 2 +9z t 3 +16z t 4

21 10.6(b) Explain the regression you would run to estimate γ h Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

22 Run the OLS estimation y t = α 0 + γ 0 (x 1t )+γ 1 (x 2t )+γ 2 (x 3t )+u t we will find ˆγ h thereafter we can find ˆδ j =ˆγ 0 +ˆγ 1 j +ˆγ 2 j 2

23 10.6(c) The Polynomial distribute lag model is a restricted version of the general model. How many restriction are imposed? How would you test these? Tutorial 2: Time Series - Regression Analysis and Further Issues (Part BEE2006: 1) Statistics and Econometrics

24 Recall that the General Model: (Unrestricted Model) y t = α 0 + δ 0 z t + δ 1 z t 1 + δ 2 z t 2 + δ 3 z t 3 + δ 4 z t 4 + u t has 6 variables and the Polynomial Model (restricted Model) y t = α 0 + γ 0 x 1t + γ 1 x 2t + γ 2 x 3t + u t only has 4 variable.

25 Simply run the restricted model and find the R 2 ur and the restricted model to find R 2 r. There are hence: Two restrictions, moving from the unrestricted to restricted model We don t have to really concern ourselves about what the restrictions might be but we know that there are two restrictions F stat = (R2 ur R2 u)/2 (1 Rur 2 )/(n 6) F 2,n 6