# How to calculate $\phi$ (phi) - a first order autocorrelation coefficient

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

• 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. – Glen_b May 7 '15 at 11:51
• – Tim Jun 27 '16 at 11:45

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.