Forecast. Forecast is the linear function with estimated coefficients. Compute with predict command

Transcription

1 Forecast Forecast is the linear function with estimated coefficients T T + h = b0 + b1timet + h Compute with predict command

2 Compute residuals Forecast Intervals eˆ t = = y y t+ h t+ h yˆ b t+ h 0 b Time 1 t Compute standard deviation of forecast These are constant over time Add to predicted values Identical to constant mean case

3 Out of Sample Forecast Out of sample prediction might be too low.

4 Out of Sample Women s Labor Force Participation No: Prediction was way too high!

5 Men s Labor Force Participation Rate

6 Estimation

7 In Sample Fit

8 Residuals

9 Forecast End of Sample looks worrying

10 Out of Sample Men s Labor Force Participation Linear Trend Terrible

11 Example 2 Retail Sales

12 Linear and Quadratic Trend

13 Linear and Quadratic Trend

14 Forecast

15 Residuals

16 Actual Values

17 Example 3: Volume

18 Estimating Logarithmic Trend

19 Fitted Trend

20 Residuals

21 Forecast

22 Out of Sample

23 Forecasting Levels from a Forecast of Logs Let Y t be a series and y t =ln(y t ) its logarithm Suppose the forecast for the log is a linear trend: E(y t+h Ω t ) = T t = β 0 + β 1 Time t Then a forecast for Y t is exp(t t ) If [L T, U T ] is a forecast interval for y T+h Then [exp(l T ), exp(u T )] is a forecast interval for Y T+h In other words, just take your point and interval forecasts, and apply the exponential function. In STATA, use generate command

24 Forecast in Levels

25 Out of Sample

26 Example 4: Real GDP

27 Ln(Real GDP)

28 Estimation

29 Fitted Trend

30 Residuals

31 Forecast of ln(rgdp)

32 Forecast of RGDP (in levels)

33 Out of Sample

34 Problems with Pure Trend Forecasts Trend forecasts understate uncertainty Actual uncertainty increases at long forecast horizons. Short term trend forecasts can be quite poor unless trend lined up correctly Long term trend forecasts are typically quite poor, as trends change over long time periods It is preferred to work with growth rates, and reconstruct levels from forecasted growth rates (more on this later).

35 Trend Models I hope I ve convinced you to be skeptical of trend based forecasting. The problem is that there is no economic theory for constant trends, and changes in the trend function are not apparent before they occur. It is better to forecast growth rates, and build levels from growth.

36 Final Trend Forecast World Record 100 meter sprint

37 Changing Trends We have seen in some cases that it appears that the trend slope has changed at some point. This is a type of structural change, sometimes called a changing trend or breaking trend. We can model this using the interaction of dummy variables with the trend.

38 Labor Force Participation Men Separate trends fit to and

39 Sub Sample Trend Lines If you fit a trend for observations before and after a breakdate τ, then for t τ and for t>τ T 1 t = β 0 + β Time t T 1 = α + α Time t 0 t Notice that both the intercept and slope change

40 Estimation You can simply estimate on each sub sample separately, and then forecast using the second set of estimates. Or, you can use dummy variable interactions. Define the dummy variable for observations after time τ d t = 1 t ( τ )

41 Dummy Equation where This is a linear regression, with regressors Time t, d t and Time t d t ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t t t t t t t t t d Time d Time t Time Time t Time t Time T β β β β τ β α β α β β τ α α τ β β = = + + < + = β α β β α β = =

42 Estimation

43 Fitted

44 Discontinuity One problem with this method is that the estimated trend function can be discontinuous At the breakdate τ there might be a jump in the trend function This might not be sensible We may wish to impose continuity In the model, this requires or β0 + β1τ = α0 + α1τ β 2 + β3τ = 0

45 Continuous Break You can impose a continuous trend by using a technique known as a spline T where t = β + β Time 0 1 ( Time τ ) ( t τ ) 0 1 t 2 t 1 = β + β Time Time t + β + β Time The variable Time t* is 0 before the breakdate, and is a smoothly increasing trend afterwards. 2 * t ( Time τ ) ( τ ) = t * t t 1

46 Fitted Continuous Trend

47 Continuous Trend Forecast

48 Contrast with Linear Trend Forecast

49 Break in 1974q1 Real GDP

50 Real GDP fitted

51 Forecast Breaking Trend Model

52 Contrast Forecast from Linear Trend

53 How to pick Breaks/Breakdates With caution, and skeptically Always have plenty of data (at least 10 years) after the breakdate Look for economic explanations Formally, the breakdate can be selected by minimizing the sum of squared errors