0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the...
Transcript of 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the...
![Page 1: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/1.jpg)
Forecast
• Forecast is the linear function with estimated coefficients
• Compute with predict command
hThT TimebbT ++ += 10
![Page 2: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/2.jpg)
Forecast Intervals
• Compute residuals
• Compute standard deviation of forecast– These are constant over time
• Add to predicted values– Identical to constant mean case
tht
hthtt
Timebbyyye
10
ˆˆ−−=
−=
+
++
![Page 3: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/3.jpg)
Out‐of‐Sample Forecast
• Out of sample prediction might be too low.
![Page 4: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/4.jpg)
Out‐of‐SampleWomen’s Labor Force Participation
• No: Prediction was way too high!
![Page 5: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/5.jpg)
Men’s Labor Force Participation Rate
![Page 6: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/6.jpg)
Estimation
![Page 7: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/7.jpg)
In‐Sample Fit
![Page 8: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/8.jpg)
Residuals
![Page 9: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/9.jpg)
Forecast
• End of Sample looks worrying
![Page 10: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/10.jpg)
Out‐of‐SampleMen’s Labor Force Participation
• Linear Trend Terrible
![Page 11: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/11.jpg)
Example 2Retail Sales
![Page 12: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/12.jpg)
Linear and Quadratic Trend
![Page 13: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/13.jpg)
Linear and Quadratic Trend
![Page 14: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/14.jpg)
Forecast
![Page 15: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/15.jpg)
Residuals
![Page 16: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/16.jpg)
Actual Values
![Page 17: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/17.jpg)
Example 3: Volume
![Page 18: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/18.jpg)
Estimating Logarithmic Trend
![Page 19: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/19.jpg)
Fitted Trend
![Page 20: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/20.jpg)
Residuals
![Page 21: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/21.jpg)
Forecast
![Page 22: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/22.jpg)
Out‐of‐Sample
![Page 23: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/23.jpg)
Forecasting Levels from a Forecast of Logs
• Let Yt be a series and yt=ln(Yt) its logarithm• Suppose the forecast for the log is a linear trend: E(yt+h | Ωt) = Tt = β0+ β1 Timet
• Then a forecast for Yt is exp(Tt)• If [LT , UT] is a forecast interval for yT+h• Then [exp(LT) , exp(UT)] is a forecast interval for YT+h
• In other words, just take your point and interval forecasts, and apply the exponential function.– In STATA, use generate command
![Page 24: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/24.jpg)
Forecast in Levels
![Page 25: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/25.jpg)
Out‐of‐Sample
![Page 26: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/26.jpg)
Example 4: Real GDP
![Page 27: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/27.jpg)
Ln(Real GDP)
![Page 28: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/28.jpg)
Estimation
![Page 29: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/29.jpg)
Fitted Trend
![Page 30: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/30.jpg)
Residuals
![Page 31: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/31.jpg)
Forecast of ln(RGDP)
![Page 32: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/32.jpg)
Forecast of RGDP (in levels)
![Page 33: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/33.jpg)
Out‐of‐Sample
![Page 34: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/34.jpg)
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).
![Page 35: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/35.jpg)
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.
![Page 36: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/36.jpg)
Final Trend ForecastWorld Record – 100 meter sprint
![Page 37: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/37.jpg)
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.
![Page 38: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/38.jpg)
Labor Force Participation ‐Men
• Separate trends fit to 1950‐1982 and 1983‐1992
![Page 39: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/39.jpg)
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
• Notice that both the intercept and slope change
tt TimeT 10 ββ +=
tt TimeT 10 αα +=
![Page 40: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/40.jpg)
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
( )τ≥= tdt 1
![Page 41: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/41.jpg)
Dummy Equation
where
• This is a linear regression, with regressorsTimet, dt and Timetdt
( ) ( ) ( ) ( )( ) ( ) ( )( ) ( )
tttt
tt
ttt
dTimedTimetTimeTime
tTimetTimeT
3210
110010
1010
111
ββββτβαβαββ
ταατββ
+++=≥−+−++=
≥++<+=
113
002
βαββαβ
−=−=
![Page 42: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/42.jpg)
Estimation
![Page 43: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/43.jpg)
Fitted
![Page 44: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/44.jpg)
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
ταατββ 1010 +=+
032 =+ τββ
![Page 45: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/45.jpg)
Continuous Break
• You can impose a continuous trend by using a technique known as a spline
where
• The variable Timet* is 0 before the breakdate, and is a smoothly increasing trend afterwards.
( ) ( )*
210
210 1
tt
ttt
TimeTime
tTimeTimeT
βββ
ττβββ
++=
≥−++=
( ) ( )ττ ≥−= tTimeTime tt 1*
![Page 46: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/46.jpg)
Fitted Continuous Trend
![Page 47: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/47.jpg)
Continuous Trend Forecast
![Page 48: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/48.jpg)
Contrast with Linear Trend Forecast
![Page 49: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/49.jpg)
Real GDP
• Break in 1974q1
![Page 50: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/50.jpg)
Real GDP ‐ fitted
![Page 51: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/51.jpg)
Forecast – Breaking Trend Model
![Page 52: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/52.jpg)
Contrast – Forecast from Linear Trend
![Page 53: 0 1 T b bTime T h - University of Wisconsin–Madisonbhansen/390/390Lecture6.pdf · • Suppose the forecast for the log is a linear trend: E(y t+h | Ω t) = T t = β 0 + β 1 Time](https://reader033.fdocuments.in/reader033/viewer/2022050413/5f89cdc1dc352536111f2f54/html5/thumbnails/53.jpg)
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