This would be a lot easier to understand if you have already taken the Regression Analysis Course.

Basically, SE(phi) = sqrt( (RSS / n-2) / sum(X* ^2) )
where n = the no. of observations (so yes, you are correct in supposing that degrees of freedom = 6 = 8-2)

RSS = the sum of the squares of the differences between Y and the fitted Y from the regression
--> Using the first observation (X = 0.44, Y = 1.05), the fitted Y is computed as follows:
fitted Y = the intercept from the regression (0.585417) + the estimated phi (0.542508) * X (0.44) = 0.82412

--> Then (Y - fitted Y)^2 = (1.05 - 0.82412)^2 = 0.051022

Do this for all observations and get the sum, which would be the RSS.

X* ^2 = (X - average of all the X's)^2
e.g. for the first observation, this would be (0.44 - 0.99)^2 = 0.3025
Then do this for all observations yet again and get the sum.



I hope this makes sense to you.

I have no idea how to manually get the standard error found by excel (.289466).  If anyone has an idea how to get this I would appreciate any help.

Note: We are using 6 degree's of freedom when calculating the t value?  That is the only way for me to match the confidence intervals using excels standard error.

I assume this is because we had 8 entries (or rather pairs of Y_t and Y_t-1) and estimated 2 parameters, correct?

Thanks for any clarification.

I have a question on how the standard deviation should be calculated. Using the formula on page 41, I calculate the standard deviation as s=((sum(Yt-phi)^2)/8)^.5. This however does not get me close to the standard deviation I get from my excel analysis (0.27476412), it got me to 0.653814. I don’t understand the assumption that the standard error is =((1-0.542508^2)/8)^0.5=0.297. Where does this formula come from?

I got phi-rat=0.542508. standard error=((1-0.542508^2)/8)^0.5=0.297

T-value for 95% and 8 freedom is equal to 2.306,

so, confidence interval is (0.542508-0.297*2.306,0.542508+0.297*2.306
)=(-0.14238,1.227398).

should we calculate the confidence intervals as stated in the HW?  i've been trying real hard but could not replicate the results from Excel as far as the confidence intervals are concerned

i'm able to estimate the phi manually and replicate Excel output...

anybody please help?!  i really like to understand so that if this same thing is asked in the Final Exam, I could answer. thanks! 

[NEAS: Cryer and Chan use z-values for confidence intervals, not t-values, to make the logic simpler. They assume the paramters are known with certainty. The proper confidence intervals use t-values. The final exam problems give you the t-value or z-value to use.]

I made the same mistake until I read the section more carefully.

It has to do with the "conditional" part of the explanation, I believe. I got the same .49 value when I calculated Y-bar using all 9 values. Based on the equation at the top of page 155, Y-bar should be the average of the first 8 Y values only. This makes sense based on the sentence right above equation 7.2.2 on page 154 too and the basic idea that we're using the Y_t_1 observations as X and Y_t observations as Y.

Doing this results in the same estimate of .54 that the Excel regression tool returns.

Following the NEAS post I entered

X as {0.44 1.05 0.62 0.72 1.08 1.24 1.42 1.35}.

and Y as {1.05 0.62 0.72 1.08 1.24 1.42 1.35 1.50}.

After I did the regression I got 0.5425 as the coeffecient for X which would be the phi value. And {-0.16579,1.25081} as the 95% confidence level for phi.

When I perform the phi hat calculation on page 155 of the text I got .4904. Is this just an estimation and that is why it does not match the answer from the regression or did I set up the regression wrong?

 

is the Y-bar equals to the average of Yt?
so i get the phi-hat equals to 0.490371934 with the equation on page 155, while excel shows the coefficient of Yt-1 is 0.614716172
by using the regression analysis
and the 95% confidence interval is [0.158126732,1.071305612]

why is the difference that large?

and what is the intercept coefficient for?
y=a+bt
i know it is stand for a for a linear regression

Doc, if I got any of the assignments incorrect, it was this one.  That being said, I converted Y_t into Y-bar_t, to create a zero-mean function of time.  I used homework 10's method to find r₁ for what I feel constitute's "regression analysis".  This is the estimate of phi-hat.

For part B I used Excel's regression to find the confidence interval, if your estimate is within the 95% confidence interval, then it is quite likely that you wouldn't reject this assumption.

RDH

We are able to use the regression feature of Excel with the model of y = a+bx. But the first-order case for AR(1) models is of the form

Y_t - mu = phi(T_{t-1} - mu) + e_t

How is mu being considered in the y = a +bx model? If b is phi, then what does a represent? Would e_t represent the RSS term?

Thanks.