Forecast with Trend Model

Transcription

1 Forecast with Trend Model The point forecast is the linear function with estimated coefficients T T + h = b0 + b1timet + h Estimate coefficients using regress Compute forecasts with predict Forecast intervals: Compute standard deviation of forecast with predict Add and subtract, multiplied by normal quantile

2 Example 1 Women s Labor Force Participation Rate Labor Force Participation Rate: 20 years and over, Women m1 1960m1 1970m1 1980m1 1990m1 time

3 Regression on Time Trend regress women time if time<tm(1990m1) predict wp if time<tm(1990m1) tsline women wp if time<tm(1990m1) Source SS df MS Number of obs = 504 F(1, 502) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = women Coef. Std. Err. t P> t [95% Conf. Interval] time _cons

4 In-Sample Fit m1 1960m1 1970m1 1980m1 1990m1 time Labor Force Participation Rate: 20 years and over, WomenFitted values

5 Residuals Residuals are difference between data and fitted regression line predict e if time<tm(1990m1), residuals eˆ t = = y y t+ h t+ h T t b 0 b Time 1 t

6 Residual Plot Residuals m1 1960m1 1970m1 1980m1 1990m1 time

7 In-Sample Fit m1 1960m1 1970m1 1980m1 1990m1 time Labor Force Participation Rate - 20 years and over, WomenFitted values wp1 wp2 predict wp if time<tm(1990m1) predict sw, stdf generate wp1 = wp *sw generate wp2 = wp *sw tsline women wp wp1 wp2 if time<tm(1990m1), lpattern (solid solid dash dash) Does the in-sample fit look good?

8 Stata do file regress women time if time<tm(1990m1) predict wp if time<tm(1990m1) predict wf if time>=tm(1990m1) predict sw, stdf generate wp1 = wf *sw generate wp2 = wf *sw generate women1=women if time<tm(1990m1) label variable women1 women tsline women1 wp wf wp1 wp2, lpattern (solid solid dash shortdash shortdash) tsline women wp wf wp1 wp2, lpattern (solid solid dash shortdash shortdash)

9 Out-of-Sample Forecast m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 time women Fitted values wf2 Fitted values wf1 Out of sample prediction might be too low.

10 Actual Out-of-Sample m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 time Labor Force Participation Rate: 20 years and over, WomenFitted values Fitted values wf1 wf2 No: Prediction was way too high!

11 Men s Labor Force Participation Rate Labor Force Participation Rate - 20 years and over, Men m1 1960m1 1970m1 1980m1 1990m1 time

12 Estimation. regress men time if time<tm(1990m1) Source SS df MS Number of obs = 504 F( 1, 502) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = men Coef. Std. Err. t P> t [95% Conf. Interval] time _cons

13 In-Sample Fit m1 1960m1 1970m1 1980m1 1990m1 time Labor Force Participation Rate - 20 years and over, Men mp1 Fitted values mp2

14 Residuals Residuals m1 1960m1 1970m1 1980m1 1990m1 time

15 Forecast m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 time men Fitted values mf2 Fitted values mf1 End of Sample looks worrying

16 Actual Out-of-Sample m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 time Civilian Labor Force Participation Rate: 20 years and over, Men Fitted values Fitted values mf1 mf2 Linear Trend Terrible

17 Example 2: Transaction Volume volume 0 1.0e e e e e+08 01jan jan jan jan jan1990 time index

18 Estimating Logarithmic Trend. regress lvolume time if time<td(07jan1994) Source SS df MS Number of obs = 2,296 F(1, 2294) = Model Prob > F = Residual , R-squared = Adj R-squared = Total , Root MSE =.2967 lvolume Coef. Std. Err. t P> t [95% Conf. Interval] time e _cons

19 Fitted Trend jan jan jan jan jan1990 time index lvolume Fitted values

20 Residuals Residuals jan jan jan jan jan1990 time index

21 Forecast jan jan jan jan jan jan jan jan2020 time index lvolume vf1 Fitted values vf2

22 Actual Out-of-Sample jan198001jan198501jan199001jan199501jan200001jan200501jan201001jan2015 time index lvolume vf1 Fitted values vf2

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 e e e e+09 01jan198001jan198501jan199001jan199501jan200001jan200501jan201001jan2015 time index volume evf1 forecast evf2

25 Actual Out-of-Sample 0 2.0e+094.0e+096.0e+098.0e+091.0e+10 01jan198001jan198501jan199001jan199501jan200001jan200501jan201001jan2015 time index volume evf1 forecast evf2

26 Stata do file use s&p gen lvolume = ln(volume) regress lvolume time if time<td(07jan1994) predict vf if time>=td(07jan1994) predict sv, stdf generate vf1 = vf *sv generate vf2 = vf *sv generate lvolume1=lvolume if time<td(07jan1994) label variable lvolume1 lvolume tsline lvolume1 vf vf1 vf2, lpattern (solid dash shortdash shortdash) tsline lvolume vf vf1 vf2 if time>=td(01jan1980), lpattern (solid dash shortdash shortdash) generate volume1=volume if time<td(07jan1994) label variable volume1 volume generate evf = exp(vf) label variable evf forecast generate evf1 = exp(vf1) generate evf2 = exp(vf2) tsline volume1 evf evf1 evf2 if time>=td(01jan1980), lpattern (solid dash shortdash shortdash) tsline volume evf evf1 evf2 if time>=td(01jan1980), lpattern (solid dash shortdash shortdash)

27 Example 3: Real GDP Real Gross Domestic Product q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time

28 Ln(Real GDP) ln_rgdp q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time

29 Estimation. regress ln_rgdp time if time<=tq(1990q4) Source SS df MS Number of obs = 176 F( 1, 174) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = ln_rgdp Coef. Std. Err. t P> t [95% Conf. Interval] time _cons

30 Fitted Trend q1 1960q1 1970q1 1980q1 1990q1 time ln_rgdp Fitted values

31 Residuals Residuals q1 1960q1 1970q1 1980q1 1990q1 time

32 Forecast of ln(rgdp) q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time ln_rgdp gf1 Fitted values gf2

33 Forecast of RGDP (in levels) q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time rgdp egf1 forecast egf2

34 Actual Out-of-Sample q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time Real Gross Domestic Product egf1 forecast egf2

35 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).

36 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.

37 Final Trend Forecast World Record 100 meter sprint year record Men Women

38 Usain Bolt

39 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.

40 Labor Force Participation - Men m1 1960m1 1970m1 1980m1 1990m1 time men Separate trends fit to and

41 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 = β 0 + β Time t T 1 T α 0 1 = α + Time t t Notice that both the intercept and slope change

42 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 )

43 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 = =

44 Estimation gen d = (time>=tm(1983m1)) gen dtime = d*time regress men time d time if time<tm(1990m1) predict p3 if time<tm(1990m1) tsline men p3 if time<tm(1990m1) m1 1960m1 1970m1 1980m1 1990m1 time men Fitted values

45 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

46 Continuous Break You can impose a continuous trend by using a technique known as a spline T where t = β 0 1 ( Time τ ) ( t τ ) 0 + β1timet + β2 t 1 = β + β Time Time t + 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

47 Fitted Continuous Breaking Trend gen stime = (time-tm(1983m1))*d regress men time stime if time<tm(1990m1) predict p4 if time<tm(1990m1) tsline men p4 if time<tm(1990m1) m1 1960m1 1970m1 1980m1 1990m1 time men Fitted values

48 Continuous Breaking Trend Forecast m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 time men Fitted values f2 Fitted values f1

49 Contrast with Linear Trend Forecast m1 1960m1 1970m1 1980m1 1990m1 2000m1 2010m1 2020m1 time men Fitted values ff2 Fitted values ff1

50 Real GDP. generate tstar=(t-tq(1974q1))*(t>=tq(1974q1)). regress y t tstar if t<=tq(1990q4) Source SS df MS Number of obs = 176 F( 2, 173) = Model Prob > F = Residual R-squared = Adj R-squared = Total Root MSE = y Coef. Std. Err. t P> t [95% Conf. Interval] t tstar _cons Break in 1974q1

51 Real GDP - fitted q1 1960q1 1970q1 1980q1 1990q1 time ln_rgdp Fitted values

52 Forecast Breaking Trend Model q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time Real Gross Domestic Product ef1 ef ef2

53 Contrast Forecast from Linear Trend q1 1960q1 1970q1 1980q1 1990q1 2000q1 2010q1 2020q1 time Real Gross Domestic Product eff1 eff eff2

54 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

55 Assignments Read Diebold through Chapter 5 Problem Set # 3 Due Tuesday (2/7) Read Chapter 3 from The Signal and the Noise Reading Reflection Due Thursday (2/9)