Introduction to Time Series Regression and Forecasting

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Chapter 14

• Introduction to Time Series Regression and
  Forecasting

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Introduction to Time Series Regression and
Forecasting(SW Chapter 14)
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Example 1 of time series data US rate of price
inflation, as measured by the quarterly
percentage change in the Consumer Price Index
(CPI), at an annual rate
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Example 2 US rate of unemployment
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Why use time series data?
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Time series data raises new technical issues
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Using Regression Models for Forecasting (SW
Section 14.1)
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Introduction to Time Series Data and Serial
Correlation (SW Section 14.2)
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We will transform time series variables using
lags, first differences, logarithms, growth
rates
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Example Quarterly rate of inflation at an annual
rate (U.S.)
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Example US CPI inflation its first lag and
its change
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Autocorrelation
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Sample autocorrelations
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Example
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Other economic time series
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Other economic time series, ctd
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Stationarity a key requirement for external
validity of time series regression
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Autoregressions(SW Section 14.3)
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The First Order Autoregressive (AR(1)) Model
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Example AR(1) model of the change in inflation
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Example AR(1) model of inflation STATA
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Example AR(1) model of inflation STATA, ctd.
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Example AR(1) model of inflation STATA, ctd
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Forecasts terminology and notation
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Forecast errors
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Example forecasting inflation using an AR(1)
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The AR(p) model using multiple lags for
forecasting
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Example AR(4) model of inflation
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Example AR(4) model of inflation STATA
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Example AR(4) model of inflation STATA, ctd.
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Digression we used ?Inf, not Inf, in the ARs.
Why?
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So why use ?Inft, not Inft?
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Time Series Regression with Additional Predictors
and the Autoregressive Distributed Lag (ADL)
Model (SW Section 14.4)
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Example inflation and unemployment
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The empirical U.S. Phillips Curve, 1962 2004
(annual)
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The empirical (backwards-looking) Phillips Curve,
ctd.
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Example dinf and unem STATA
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Example ADL(4,4) model of inflation STATA,
ctd.
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The test of the joint hypothesis that none of the
Xs is a useful predictor, above and beyond
lagged values of Y, is called a Granger causality
test
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Forecast uncertainty and forecast intervals
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The mean squared forecast error (MSFE) is,
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The root mean squared forecast error (RMSFE)
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Three ways to estimate the RMSFE
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The method of pseudo out-of-sample forecasting
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Using the RMSFE to construct forecast intervals
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Example 1 the Bank of England Fan Chart,
11/05
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Example 2 Monthly Bulletin of the European
Central Bank, Dec. 2005, Staff macroeconomic
projections
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Example 3 Fed, Semiannual Report to Congress,
7/04
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Lag Length Selection Using Information Criteria
(SW Section 14.5)
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The Bayes Information Criterion (BIC)
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Another information criterion Akaike Information
Criterion (AIC)
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Example AR model of inflation, lags 06
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Generalization of BIC to multivariate (ADL)
models
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Nonstationarity I Trends(SW Section 14.6)
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Outline of discussion of trends in time series
data
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1. What is a trend?
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What is a trend, ctd.
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Deterministic and stochastic trends
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Deterministic and stochastic trends, ctd.
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Deterministic and stochastic trends, ctd.
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Deterministic and stochastic trends, ctd.
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Stochastic trends and unit autoregressive roots
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Unit roots in an AR(2)
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Unit roots in an AR(2), ctd.
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Unit roots in the AR(p) model
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Unit roots in the AR(p) model, ctd.
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2. What problems are caused by trends?
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Log Japan gdp (smooth line) and US inflation
(both rescaled), 1965-1981
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Log Japan gdp (smooth line) and US inflation
(both rescaled), 1982-1999
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3. How do you detect trends?
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DF test in AR(1), ctd.
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Table of DF critical values
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The Dickey-Fuller test in an AR(p)
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When should you include a time trend in the DF
test?
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Example Does U.S. inflation have a unit root?
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Example Does U.S. inflation have a unit root?
ctd
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DF t-statstic 2.69 (intercept-only)
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4. How to address and mitigate problems raised
by trends
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Summary detecting and addressing stochastic
trends
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Nonstationarity II Breaks and Model Stability
(SW Section 14.7)
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A. Tests for a break (change) in regression
coefficients
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Case II The break date is unknown
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The Quandt Likelihod Ratio (QLR) Statistic (also
called the sup-Wald statistic)
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The QLR test, ctd.
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Has the postwar U.S. Phillips Curve been stable?
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QLR tests of the stability of the U.S. Phillips
curve.
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B. Assessing Model Stability using Pseudo
Out-of-Sample Forecasts
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Application to the U.S. Phillips Curve
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POOS forecasts of ?Inf using ADL(4,4) model with
Unemp
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poos forecasts using the Phillips curve, ctd.
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Summary Time Series Forecasting Models (SW
Section 14.8)
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Summary, ctd.