Category Using gret l for Principles of Econometrics, 4th Edition

Time-Varying Volatility and ARCH Models: Introduction to Financial Econometrics

In this chapter we’ll estimate several models in which the variance of the dependent variable changes over time. These are broadly referred to as ARCH (autoregressive conditional heteroskedas – ticity) models and there are many variations upon the theme.

The first thing to do is illustrate the problem graphically using data on stock returns. The data are stored in the gretl dataset returns. gdt. The data contain four monthly stock price indices: U. S. Nasdaq (nasdaq), the Australian All Ordinaries (allords), the Japanese Nikkei (nikkei) and the U. K. FTSE (ftse). The data are recorded monthly beginning in 1988:01 and ending in 2009:07. Notice that with monthly data, the suffix is two digits, that is 1988:01 is January (01) in the year 1988.

Simple scatter plots appear below...

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Two nonstationary series are cointegrated if they tend to move together through time. For instance, we have established that the levels of the Fed Funds rate and the 3-year bond are non­stationary, whereas their differences are stationary. In the opaque language used in time-series
literature, each series is said to be “integrated of order 1” or I(1). If the two nonstationary series move together through time then we say they are “cointegrated.” Economic theory would suggest that they should be tied together via arbitrage, but that is no guarantee. In this context, testing for cointegration amounts to a test of the substitutability of these assets.

The basic test is very simple. Regress one I(1) variable on another using least squares...

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The tobit model is essentially just a linear regression where some of the observations on your dependent variable have been censored. A censored variable is one that, once it reaches a limit, it is recorded at that limit no matter what it’s actual value might be. For instance, anyone earning $1 million or more per year might be recorded in your dataset at the upper limit of $1 million. That means that Bill Gates and the authors of your textbook earn the same amount in the eyes of your dataset (just kidding, gang). Least squares can be seriously biased in this case and it is wise to use a censored regression model (tobit) to estimate the parameters of the regression when a portion of your sample is censored.

Hill et al. (2011) consider the following model of hours worked for a sample of w...

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The basic ARCH(1) model can be expressed as:

yt = в + et


et|1t-i ~ N(0, ht)


ht = ao + aqe2_i


a0 > 0, 0 < a1 < 1

The first equation describes the behavior of the mean of your time-series. In this case, equation (14.1) indicates that we expect the time-series to vary randomly about its mean, в. If the mean of your time-series drifts over time or is explained by other variables, you’d add them to this equation just as you would a regular regression model. The second equation indicates that the error of the regression, et, are normally distributed and heteroskedastic. The variance of the current period’s error depends on information that is revealed in the preceding period, i. e., It_i. The variance of et is given the symbol ht...

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A “Secondary Section” is a named appendix or a fr...

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Simultaneous Equations Models

In Chapter 11 of POE4 the authors present a model of supply and demand. The econometric model contains two equations and two dependent variables. The distinguishing factor for models of this type is that the values of two (or more) of the variables are jointly determined. This means that a change in one of the variables causes the other to change and vice versa. The estimation of a simultaneous equations model is demonstrated using the truffle example which is explained below.

11.1 Truffle Example

Consider a supply and demand model for truffles:

qi =ai + a2pi + a3psi + a^dii + ed (11.1)

qi =ві + в Pi + вз pfi + es (11.2)

The first equation (11.1) is demand and q us the quantity of truffles traded in a particular French market, p is the market price of truffles, ps is the market price ...

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