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I have a dataset of historical quarterly earnings per share for 8 years. I am trying to use the following formula for the purpose of estimating earnings: $E(Q_t) =Q_{t-4} + \phi_1(Q_{t-1} - Q_{t-5}) + δ$; where $\phi_1$ is given by the first order autocorrelation coefficient ($r_1$). $E(Q_t)$ is the expected quarterly earnings for quarter $t$.

Can anyone explain to me how to calculate $\phi_1$?

It is important that the calculation is based only on the historical data available.

P.S. The formula appears on page 5 of the following article: "Quarterly Accounting Data: Time-Series Properties and Predictive-Ability Results" by George Foster The Accounting Review Vol. 52, No. 1 (Jan., 1977), pp. 1-21

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    $\begingroup$ Unless you have the population, you can't calculate a population parameter. You can either estimate $\phi$ or you can calculate $\hat{\phi}$. Please also give a complete reference to the article you mention. $\endgroup$ – Glen_b May 7 '15 at 11:51
  • $\begingroup$ Check stats.stackexchange.com/questions/185521/… $\endgroup$ – Tim Jun 27 '16 at 11:45
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I am assuming you're not going to calculate this thing manually. Use a package such as R with arima class.

If you really are to do this manually in Excel, then here's one way. Construct a new dependent variable $Y=Q_t-Q_{t-4}$, and the independent variable $X=Q_{t-1}-Q_{t-5}$. You get a new model: $$Y_t=X_t+\delta_t$$ Next, use the regression through origin to find: $$\phi=\frac{\sum YX}{\sum X^2}$$

You can see by the construction of $Y$ and $X$ that they're using overlapping intervals, basically YoY changes in earnings at quarterly frequency. This creates an issue of autocorrelation. Also, the model doesn't have an intercept, which is problematic too.

So, I would not use this model to make money, it's not going to work.

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