About This Book
Purpose
The purpose of this book is to show how broadly the VARMAX procedure supports modern time series econometrics. The VARMAX procedure includes modern facilities like automatic model selection and GARCH models for univariate series. But the main focus is on multivariate time series, for which automatic VARMA model selection and GARCH are of course supported. Moreover, BVAR models, together with subjects like Granger Causality and cointegration, are supported. All these featured are illustrated mainly by examples using real data.
Is This Book for You?
This book is useful for readers who are analyzing a time series for the first time. They will find PROC VARMAX easy to use. But PROC VARMAX also includes many advanced features; therefore, readers who know more advanced theoretical time series models will find this book useful as a guide for applying PROC VARMAX for advanced model building.
Prerequisites
The book is aimed at econometricians who have completed at least one course in time series modeling.
Scope of This Book
Chapters 2 through 4 give the background for time series models as a special case of regression analysis. In these chapters, you will learn how ordinary regression fails; for example, see Figure 1.2. Chapters 2 through 4 also demonstrate how these failures to some extent can be accounted for. These methods are, however, not sufficient to establish reliable statistical models for many common data problems.
The models focused on are models for multivariate time series—that is, models for the interdependence of two or more univariate time series. Such models can be seen as generalizations of the usual regression model to the case of multivariate, left side, response variables. Relationships among time series are not necessarily immediate but can happen with some time delay. In order to model such delays, both wages and prices have to be right side variables in a regression model with both wages and prices as right side variables. In time series, a system like this one is said to “have feedback.” A major part of the book is devoted to describing such models and to showing by example how you can do the analysis by means of the VARMAX procedure (Chapters 7 through 12).
Another assumption underlying the usual regression model is often violated. The variance in many situations is nonconstant, so that the residuals cannot be identically distributed. One simple example is that the variance often increases as the level increases. In many situations, this problem is rather easily solved by a logarithmic transformation. In more detailed analysis, this transformation can be refined by a Box-Cox transformation. This topic is, however, beyond the scope of this book.
For time series, the variance can vary in a seemingly random manner even if the variance is constant in a broad sense. A typical example is a stock rate that for some days is very volatile but in other periods is nearly constant. For such series, the variance can be considered as a time series in itself, which can be modeled by the so-called GARCH models. These methods are also covered by PROC VARMAX. See Chapters 15 through 17.
In modern econometric analysis of time series data, cointegration and error correction models play a major role. The basic idea is that, even if two or more time series seem to be unstable individually, some stable relationship exists among them. This stable relationship can be considered as an economic equilibrium. In this case, the series are said to be cointegrated. If the series for some reason are away from this stable relationship, an error correction mechanism can describe how they find their way back to equilibrium. So dynamics of economic data can be modeled in a way that is closely related to economic theory. Similar models are useful for time series from branches other than economics. These topics are covered by Chapters 13 through 14.
About the Examples
Software Used to Develop the Book’s Content
The software used to develop the content of this book is as follows:
• SAS/STAT 14.1
• SAS/ETS 14.1
But most of the content is also available in SAS ETS 13.1.
Data Sets Used in the Book
All series are downloaded by the author at some specific point in time, so subsequent revisions of the series are, of course, not incorporated in the examples. The focus is on applications and not on specific conclusions about the series and their impact. Intuitive arguments for understanding the models based on the nature of the series are, of course, used. Otherwise, the series are analyzed without any political or economic viewpoints, to ensure that the presentation is neutral and purely technical.
Time series examples, by their very nature, soon become obsolete. Even forecasting experiments, in which more recent observations are compared with forecasts, begin to seem like historical exercises after a while. Keeping this in mind, know that forecasts in this book are in no way suggested to be the future realizations of the time series.
You can access the data, as well as example code, for this book by linking to its author’s page at http://support.sas.com/publishing/authors. Select the name of the author. Then look for the cover thumbnail of this book, and select Example Code and Data to display the SAS programs that are included in this book.
If you are unable to access the code through the Web site, send e-mail to [email protected].
WAGEPRICE
This data set includes yearly index numbers for the wage and the prices in Denmark for the years 1818–1981. It gives a total of 164 observations. The observations are taken from a small book on historical data for Denmark (Gammelgaard 1985), but originally they were published in many historical sources.
EGG
The data set includes 144 monthly observations of index numbers for the Danish-produced quantity of eggs and the price to the farmers for eggs. The data is rather old, 1965–1976, but at that time the Danish market was rather closed to foreign competition. So the relation between produced quantity and the price can be modeled without corrections for other variables. The data is published by Statistics Denmark.
QUARTERLY_MILK
The data set includes quarterly observations of the number of cows and the milk production in the United States. The data set includes observations from 1998 to 2012, a total of 60 observations. The series is quoted from an Excel data sheet found on the U.S. Department of Agriculture’s Economic Research Service website.
QUOTES
The data set includes daily observations of quotes for two stocks at the Danish stock exchange from March 21, 2002, to March 19, 2003. One firm is a bank, and the other operates in the field of biotech. Both companies have changed since the time of the observations, so firm-specific information is of no longer of interest. The series has 248 observations of the quotes, the log-transformed quotes, and the daily change in the notation for both companies expressed as a percentage.
POTATOES_YEAR
This data set includes yearly observations of the average price of potatoes in states in the United States: Delaware, Maryland, Ohio, Virginia, and Pennsylvania. The observation period is 1866 and up to 2013, giving a total of 148 observations.
The original price is the total value of the production of potatoes within the state divided by the produced quantity. The unit of the price is US Dollar per CWT (approximately 45 kg), but the precise unit of measurement is of no importance because of the transformation by logarithms.
The time series are published by United States Department of Agriculture, National Agricultural Statistics Service.
SAS University Edition
If you are using SAS University Edition to access data and run your programs, then please check the SAS University Edition page to ensure that the software contains the product or products that you need to run the code: http://support.sas.com/software/products/university-edition/index.html.
PROC VARMAX is not supported by SAS University Edition it the version available in autumn 2015, when this book was produced.
Output and Graphics Used in This Book
The output tables and the output graphics are mainly created by PROC VARMAX, which produces a huge amount of graphical output. A few figures are, however, created by PROC SGLOT. The actual code for the displayed output is included in the text and in the code at http://support.sas.com/publishing/authors/milhoj.html.
Additional Help
Although this book illustrates many analyses regularly performed in businesses across industries, questions specific to your aims and issues may arise. To fully support you, SAS Institute and SAS Press offer you the following help resources:
• For questions about topics covered in this book, contact the author through SAS Press:
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• SAS Institute maintains a comprehensive Web site with up-to-date information. One page that is particularly useful to both the novice and the seasoned SAS user is its Knowledge Base. Search for relevant notes in the “Samples and SAS Notes” section of the Knowledge Base at http://support.sas.com/resources.
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