Financial Econometrics Review Session Notes 3

Transcription

1 Financial Econometrics Review Session Notes 3 Nina Boyarchenko January 22, 2010 Contents 1 k-step ahead forecast and forecast errors Example 1: stationary series Example 2: non-stationary series Example 3: ing from an equation Augmented Dickey Fuller test Example 3 (cont.)

2 1 k-step ahead forecast and forecast errors Consider first an AR(1) model of the form: y t = β 0 + β 1 y t 1 + ε t ; ε t N(0, σ 2 ). Recall from your teaching notes that the k step ahead forecast is given by: { y f β t+k E [y k t+k y t ] = 1 y t + (1 β1 k )μ; β 1 < 1 kβ 0 + y t ; β 1 = 1 where μ = β 0 /(1 β 1 ) is the unconditional mean. The forecast error is then given by: { ( ) ( ) 1 β 2k var ε f t+k var y t+k y f t+k y 1 σ t = 2 ; β 1 β1 2 1 < 1 kσ 2 ; β 1 = 1 and the 95% confidence interval by: ( ) y f t+k var ± 2 ε f t+k. 1.1 Example 1: stationary series Let s revisit the AR(1) model in Example 3 of Review 1: y t = y t 1 + ε t ; ε t i.i.d. N(0, 1). Question 1. Find the k step ahead forecast as a function of y t and k Substituting β 1 into the above formula, we have: y f t+k = 0.75k y t + 2( k ) Consider now actually calculating the forecast through 15 periods in EViews. open a new page in the EViews workfile and choose Unstructured/Undated from the Workfile structure type pull-down menu. In the Observations: box, enter 16. The easiest way to create the forecast, is to create a program file. From the File menu, select New and then Program. Enter the following code in the program file, save it and click Run: series example1f=0.5 example1f(!i)=0.75ˆ!i*example1f(1)+2*(1-0.75ˆ!i) next 2

3 Figure 1: and forecast standard error of y t = y t 1 + ε t Notice that we are using a for loop in the program. The forecast plot is presented in Fig. 1. Consider now the 95% confidence interval. To compute it, in your program, also enter: series example1std=0 example1std(!i)=(1-0.75ˆ!i)/(1-0.75ˆ2) next series example1bp=example1f+2*@sqrt(example1std) series example1bm=example1f-2*@sqrt(example1std) 1.2 Example 2: non-stationary series Consider now another AR(1) model: y t = y t 1 + ε t ; ε t i.i.d. N(0, 1). Question 2. Find the k step ahead forecast as a function of y t and k 3

4 Substituting β 1 into the above formula, we have: y f t+k = y t + k 0.5 Consider now actually calculating the forecast through 15 periods in EViews. Enter the following code in the program file, save it and click Run: series example2f=0.5 example2f(!i)=example2f(1)+!i*0.5 next The forecast plot is presented in Fig. 2. Consider now the 95% confidence interval. To Figure 2: and forecast standard error of y t = y t 1 + ε t compute it, in your program, also enter: series example2std=0 example2std(!i)=!i 4

5 next series series Question 3. What are the differences between this graph and the previous graph? 1.3 Example 3: ing from an equation Consider the housing data from Homework 1. Begin by loading it into the EViews environment and estimating an ar(1) model for the raw returns. To compute the forecast, open once again the associated Equation object and select the forecast tab. This time, select the dynamic forecast option. This creates the graph in Fig. 3. Figure 3: and forecast standard error of housing raw returns : RAW_RETURNF Actual: RAW_RETURN sample: 1987M M10 Adjusted sample: 1987M M10 Included observations: 272 Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error Theil Inequality Coefficient Bias Proportion Variance Proportion Covariance Proportion S.E. Consider now forecasting outside the sample. To do this, first extend the range of the workfile page by double-clicking on the Range at the top of the page and changing it to the desired range (in the example, the range is increased to 2011m1, so that we have 15 forecasting periods). Then, once again open the Equation object and select the forecast tab. To just compute the out-of-sample forecast, change the range to 2009m m1. This creates the forecast in Fig. 4. 5

6 Figure 4: Out-of-sample forecast and forecast standard error of housing raw returns : RAW_RETURNF Actual: RAW_RETURN sample: 2009M M01 Included observations: 1 Root Mean Squared Error Mean Absolute Error Mean Abs. Percent Error IV I II III IV I S.E. 2 Augmented Dickey Fuller test Recall that the Dickey-Fuller test checks whether the time-series has a unit-root (i.e. is non-stationary). The test is very easy to perform in EViews. All you need to do is to open the series in question and under the View tab, select Unit root test... For the Dickey-Fuller test, you can now just click the OK button. 2.1 Example 3 (cont.) Let s perform the Dickey-Fuller test on the raw housing returns. Open the raw return series and under the View tab, select Unit root test... For the Dickey-Fuller test, you can now just click the OK button. The results of the test are reported in Table 1. Question 4. Can we reject the unit-root hypothesis? 6

7 Table 1: Dickey-Fuller test for raw housing returns Null Hypothesis: RAW RETURN has a unit root Exogenous: Constant Lag Length: 11 (Automatic - based on SIC, maxlag=15) t-statistic Prob. Augmented Dickey-Fuller test statistic Test critical values: 1% level % level % level *MacKinnon (1996) one-sided p-values. 7