1

Forecast uncertainty and forecast intervals

2

Mean Squared Forecast Error

3

Three ways to estimate the RMSFE

4

Pseudo out-of-sample forecasting

5

Constructing forecast intervals

6

Example #1: the Bank of England “Fan Chart”, 11/05

7

Example #2: Monthly Bulletin of the European Central Bank, Dec. 2005, Staff macroeconomic projections

8

Example #3: Fed, Semiannual Report to Congress, 7/04

9

Lag Length Selection Using Information Criteria

10

Bayes Information Criterion (BIC)

11

Akaike Information Criterion (AIC)

12

Example: AR model of inflation

13

Generalization of BIC to multivariate (ADL) models

14

Nonstationarity from Trends

15

1. What is a trend?

16

17

18

Deterministic and stochastic trends

19

Deterministic and stochastic trends

20

Deterministic and stochastic trends

21

Deterministic and stochastic trends

22

Stochastic trends and unit roots

23

Unit roots in an AR(2)

24

Unit roots in an AR(2), ctd.

25

Unit roots in the AR(p) model

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Unit roots in the AR(p) model, ctd.

27

2. What problems are caused by trends?

28

Log Japan gdp (smooth line) and US inflation (both rescaled), 1965-1981

29

Log Japan gdp (smooth line) and US inflation (both rescaled), 1982-1999

30

3. How do you detect trends?

31

DF test in AR(1), ctd.

32

Table of DF critical values

33

The Dickey-Fuller test in an AR(p)

34

When should you include a time trend in the DF test?

35

Example: Does U.S. inflation have a unit root?

36

Example: Does U.S. inflation have a unit root?

37

DF t-statstic = –2.69 (intercept-only):

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4. How to address and mitigate problems raised by trends

39

Summary: detecting and addressing stochastic trends

40

Nonstationarity from breaks (changes) in regression coefficients

41

42

Case II: The break date is unknown

43

The Quandt Likelihod Ratio (QLR) Statistic (also called the “sup-Wald” statistic)

44

The QLR test, ctd.

45

46

47

Has the postwar U.S. Phillips Curve been stable?

48

QLR tests of the stability of the U.S. Phillips curve.

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50

Assessing Model Stability using Pseudo Out-of-Sample Forecasts

51

Application: U.S. Phillips Curve

52

POOS forecasts of Inf using ADL(4,4) model with Unemp

53

POOS forecasts of Inf using ADL(4,4) model with Unemp