Well, that is inconvenient: the minimum and maximum temperatures are combined in a string variable. If we wanted to do some statistical analyses with these data, we would need to extract these values from the string and store t The strpos() function locates the position of one string in another string. We use the location of the “” character in the substr() function to extract the relevant portion of the string, then we apply the real() function to convert the string of digits to a number. Not difficult, but you have to know how
Introduction to Time Series Using Stata Revised Edition SEAN BECKETTI ® A Stata Press Publication StataCorp LLC College Station, Texas ® Copyright © 2013, 2020 by StataCorp LLC All rights reserved First edition 2013 Revised edition 2020 Published by Stata Press, 4905 Lakeway Drive, College Station, Texas 77845 Typeset in LATEX 2 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Print ISBN-10: 1-59718-306-7 Print ISBN-13: 978-1-59718-306-2 ePub ISBN-10: 1-59718-307-5 ePub ISBN-13: 978-1-59718-307-9 Mobi ISBN-10: 1-59718-308-3 Mobi ISBN-13: 978-1-59718-308-6 Library of Congress Control Number: 2020932011 No part of this book may be reproduced, stored in a retrieval system, or transcribed, in any form or by any means—electronic, mechanical, photocopy, recording, or otherwise—without the prior written permission of StataCorp LLC Stata, , Stata Press, Mata, , and NetCourse are registered trademarks of StataCorp LLC Stata and Stata Press are registered trademarks with the World Intellectual Property Organization of the United Nations NetCourseNow is a trademark of StataCorp LLC LATEX 2 is a trademark of the American Mathematical Society Contents Preface Acknowledgments 1 Just enough Stata 1.1 Getting started 1.1.1 Action first, explanation later 1.1.2 Now some explanation 1.1.3 Navigating the interface 1.1.4 The gestalt of Stata 1.1.5 The parts of Stata speech 1.2 All about data 1.3 Looking at data 1.4 Statistics 1.4.1 Basics 1.4.2 Estimation 1.5 Odds and ends 1.6 Making a date 1.6.1 How to look good 1.6.2 Transformers 1.7 Typing dates and date variables 1.8 Looking ahead 2 Just enough statistics 2.1 Random variables and their moments 2.2 Hypothesis tests 2.3 Linear regression 2.3.1 Ordinary least squares 2.3.2 Instrumental variables 2.3.3 FGLS 2.4 Multiple-equation models 2.5 Time series 2.5.1 White noise, autocorrelation, and stationarity 2.5.2 ARMA models 3 Filtering time-series data 3.1 Preparing to analyze a time series 3.1.1 Questions for all types of data How are the variables defined? What is the relationship between the data and the phenomenon of interest? Who compiled the data? What processes generated the data? 3.1.2 Questions specifically for time-series data What is the frequency of measurement? Are the data seasonally adjusted? Are the data revised? 3.2 The four components of a time series Trend Cycle Seasonal 3.3 Some simple filters 3.3.1 Smoothing a trend 3.3.2 Smoothing a cycle 3.3.3 Smoothing a seasonal pattern 3.3.4 Smoothing real data 3.4 Additional filters 3.4.1 ma: Weighted moving averages 3.4.2 EWMAs exponential: EWMAs dexponential: Double-exponential moving averages 3.4.3 Holt–Winters smoothers hwinters: Holt–Winters smoothers without a seasonal component shwinters: Holt–Winters smoothers including a seasonal component 3.5 Points to remember 4 A first pass at forecasting 4.1 Forecast fundamentals 4.1.1 Types of forecasts 4.1.2 Measuring the quality of a forecast 4.1.3 Elements of a forecast 4.2 Filters that forecast 4.2.1 Forecasts based on EWMAs 4.2.2 Forecasting a trending series with a seasonal component 4.3 Points to remember 4.4 Looking ahead 5 Autocorrelated disturbances 5.1 Autocorrelation 5.1.1 Example: Mortgage rates 5.2 Regression models with autocorrelated disturbances 5.2.1 First-order autocorrelation 5.2.2 Example: Mortgage rates (cont.) 5.3 Testing for autocorrelation 5.3.1 Other tests 5.4 Estimation with first-order autocorrelated data 5.4.1 Model 1: Strictly exogenous regressors and autocorrelated disturbances The OLS strategy The transformation strategy The FGLS strategy Comparison of estimates of model 1 5.4.2 Model 2: A lagged dependent variable and i.i.d errors 5.4.3 Model 3: A lagged dependent variable with AR(1) errors The transformation strategy The IV strategy 5.5 Estimating the mortgage rate equation 5.6 Points to remember 6 Univariate time-series models 6.1 The general linear process 6.2 Lag polynomials: Notation or prestidigitation? 6.3 The ARMA model 6.4 Stationarity and invertibility 6.5 What can ARMA models do? 6.6 Points to remember 6.7 Looking ahead 7 Modeling a real-world time series 7.1 Getting ready to model a time series 7.2 The Box–Jenkins approach 7.3 Specifying an ARMA model 7.3.1 Step 1: Induce stationarity (ARMA becomes ARIMA) 7.3.2 Step 2: Mind your p’s and q’s 7.4 Estimation 7.5 Looking for trouble: Model diagnostic checking 7.5.1 Overfitting 7.5.2 Tests of the residuals 7.6 Forecasting with ARIMA models 7.7 Comparing forecasts 7.8 Points to remember 7.9 What have we learned so far? 7.10 Looking ahead 8 Time-varying volatility 8.1 Examples of time-varying volatility 8.2 ARCH: A model of time-varying volatility 8.3 Extensions to the ARCH model 8.3.1 GARCH: Limiting the order of the model 8.3.2 Other extensions Asymmetric responses to “news” Variations in volatility affect the mean of the observable series Nonnormal errors Odds and ends 8.4 Points to remember 9 Models of multiple time series 9.1 Vector autoregressions 9.1.1 Three types of VARs 9.2 A VAR of the U.S macroeconomy 9.2.1 Using Stata to estimate a reduced-form VAR 9.2.2 Testing a VAR for stationarity Other tests 9.2.3 Forecasting Evaluating a VAR forecast 9.3 Who’s on first? 9.3.1 Cross correlations 9.3.2 Summarizing temporal relationships in a VAR Granger causality How to impose order FEVDs Using Stata to calculate IRFs and FEVDs 9.4 SVARs 9.4.1 Examples of a short-run SVAR 9.4.2 Examples of a long-run SVAR 9.5 Points to remember 9.6 Looking ahead 10 Models of nonstationary time series 10.1 Trends and unit roots 10.2 Testing for unit roots 10.3 Cointegration: Looking for a long-term relationship 10.4 Cointegrating relationships and VECMs 10.4.1 Deterministic components in the VECM 10.5 From intuition to VECM: An example Step 1: Confirm the unit root Step 2: Identify the number of lags Step 3: Identify the number of cointegrating relationships Step 4: Fit a VECM Step 5: Test for stability and white-noise residuals Step 6: Review the model implications for reasonableness 10.6 Points to remember 10.7 Looking ahead 11 Closing observations 11.1 Making sense of it all 11.2 What did we miss? 11.2.1 Advanced time-series topics 11.2.2 Additional Stata time-series features Data management tools and utilities Univariate models Multivariate models 11.3 Farewell References Author index Subject index