# Estimation of $\phi$ in $AR(1)$ process

Let $n=144$ observations from an $AR(1)$ model $$y_t=\phi y_{t-1}+\epsilon_t$$ where $\epsilon_t$ is White Noise with mean zero and variance $\sigma^2$. If $y_1=-1.7$, $y_{144}=-2.1$, $\sum_{t=2}^n y_t y_{t-1}=-128.6$ and $\sum_{t=1}^n y_t^2=246.4$.

Find an estimate for $\phi$.

I already found that the estimate of $\phi$ is the first sample autocorrelation Time Series Analysis: With Applications in R $$\hat{\phi}=\frac{\sum_{t=2}^n (y_t-\overline{y})(y_{t-1}-\overline{y})}{\sum_{t=1}^n (y_{t-1}-\overline{y})^2}$$

The problem is that I don't know how to evaluate it just with the information that I have.

$$\hat{\phi}=\frac{\sum_{t=2}^n (y_t y_{t-1}-\overline{y}y_{t}+\overline{y}y_{t-1}+\overline{y}^2)}{\sum_{t=1}^n (y_t^2-2\overline{y}y_{t-1}+\overline{y}^2)}$$

I have separated the sums but it seems to me that the information is insufficient. What I'm missing?

• Hint: OLS estimation minimizes $\sum(y_t-\phi y_{t-1})^2$. Take the first derivative of that with respect to $\phi$ and set it equal to zero, and you will understand why the formula you posted is not the appropriate one, for this particular specification. Careful with the indices. – Alecos Papadopoulos Dec 30 '16 at 13:38
• @AlecosPapadopoulos In this case $\overline{y}=0$ and you have $\hat{\phi}=\frac{\sum_{t=2}^n y_t y_{t-1}}{\sum_{t=1}^n y_t^2}$. This formula that I posted is just for nonzero mean process right? – user72621 Dec 30 '16 at 13:42
• Careful with the indices. What is the index run in the objective function to be minimized? – Alecos Papadopoulos Dec 30 '16 at 13:48
• Yes the general formula is for an $AR(1)$ with drift (and so non-zero mean). – Alecos Papadopoulos Dec 30 '16 at 13:49
• @MichaelChernick I provided the OP with enough hints to solve this. Then exactly because it is a self study and "our hands are tied" as regards posting an answer, I suggested that he should offer the community a service by posting himself the answer and accept it. – Alecos Papadopoulos Dec 30 '16 at 14:16

Following what @Alecos said

The OLS estimation minimizes $$\sum_{t=2}^n (y_t-\phi y_{t-1})^2=\sum_{t=2}^n (y_t^2-2\phi y_t y_{t-1}+\phi^2 y_{t-1}^2)$$ Taking derivative

$$\frac{\partial}{\partial \phi}\sum_{t=2}^n (y_t^2-2\phi y_t y_{t-1}+\phi^2 y_{t-1}^2)=-2\sum_{t=2}^n y_t y_{t-1}+2\phi\sum_{t=2}^n y_{t-1}^2=0$$ Then $$\hat{\phi}=\frac{\sum_{t=2}^n y_t y_{t-1}}{\sum_{t=2}^ny_{t-1}^2}$$

In this way the denonimator goes from $1$ to $(n-1)$, so theoretically I have to take $$\sum_{t=1}^n y_t^2-y_{144}^2=246.4-(-2.1)^2=241.99$$

Then $$\hat{\phi}=\frac{-128.6}{241.99}\approx 0.53$$

• Indeed. The question is why they provided you also with the value of $y_1$... perhaps as a little trap (that I could not resist applying it in my related previous comment!). – Alecos Papadopoulos Dec 30 '16 at 14:28
• @AlecosPapadopoulos This question has another item involving covariance in prediction errors, maybe this is what I'm going to use, but I'll try first before posting. – user72621 Dec 30 '16 at 14:31
• @Roland Maybe that should be a separate question for you to pose. Are you satisfied with your answer here? – Michael R. Chernick Dec 30 '16 at 15:53
• @MichaelChernick I do not know what you mean by "satisfied", I just want to learn how to resolve the question, however I do not see other possible solutions based on what was provided, but every suggestion is welcome. – user72621 Dec 30 '16 at 17:17
• It appears to me that you might be using the site to confirm your answer to a homework or quiz problem. The question is so specific that it is hard to believe that there is a point for additional suggestions. – Michael R. Chernick Dec 30 '16 at 21:22