## From regression to dynamic linear model: GLD, NEM, and GSPC

June 5, 2014 by zulfahmed

One of the more annoying issues for analytics of futures is the fact that the futures contracts have to be rolled. Fortunately, the gold ETF GLD has continuous returns. Then there are some strong and well-known empirical relations between the commodity and gold mining companies as well as the stock market. We can obtain the data using the quantmod library in R and obtain a basic regression relation (these are contemporaneous data):

lm(formula = GLD.R$GLD.Close ~ NEM.R$NEM.Close + GSPC.R$GSPC.Close +

DJPRE1$DBAA)

Residuals:

Min 1Q Median 3Q Max

-0.070574 -0.004731 0.000132 0.005388 0.063796

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 0.0008786 0.0005083 1.728 0.0841 .

NEM.R$NEM.Close 0.3772582 0.0088180 42.783 <2e-16 ***

GSPC.R$GSPC.Close -0.1871526 0.0163548 -11.443 <2e-16 ***

DJPRE1$DBAA -0.0002857 0.0003686 -0.775 0.4384

—

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.009365 on 1850 degrees of freedom

(13 observations deleted due to missingness)

Multiple R-squared: 0.4996, Adjusted R-squared: 0.4988

F-statistic: 615.7 on 3 and 1850 DF, p-value: < 2.2e-16

So we can see clearly some strong relations, with an R-squared close to 0.5. A natural question is what happens when we consider a dynamic linear regression model instead. The general form of such models is:

It is useful to consider what might be a reasonable way to consider the generalization to the linear regression model above. An obvious sort of choice might be to set where and is computed from the linear regression. Then we might just set and . A bit of thought shows that this will yield something equivalent to the linear regression model. So we might next consider a slightly more interesting extension and insert a stochastic volatility factor into .

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