Regression of data whose DGP includes trend

[T_FIELD]

Dear all,

Could you give me any advice?

Suppose that

(1) x(t) rejects DFtest with constant and trend and the trend is significant.
(2) y(t) rejects DFtest with constant and trend and the trend is insignificant.
(3) y(t) also rejects DFtest with constant.

When we regress y(t) on X(t), is it correct to estimate y(t)=a+b*X(t)+u(t)?
I worry about that y(t)=a+b*X(t)+c*trend+u(t) is correct, since the DGP of x(t)
contains the trend term.

I am looking forward to any advice.

Sincerely Yours,

T_FIELD

[TomDoan]

If X has a trend and Y doesn't, then Y=a+bX will force the b coefficient to zero, so you're correct that you have to include the trend in the regression in order for it to make sense as a cointegrating relation.

[T_FIELD]

Dear Dr. Tom Doan,

Thank you so much for your reply.

>you're correct that you have to include the trend in the regression in order for it to make sense as a cointegrating relation.

Can I ask some questions on your reply?

(1) I supposed x(t) and y(t) are stationary. (I supposed both variables reject DFtest).
Q1: Even so, do I have to care about cointegration? If so, could you tell me the reason or any (easy) text book?
Q2: When x(t) and y(t) are stationary, dose your answer change?
(What I want to know is, of course, the true value of parameter b. So, on the one hand, I think we need not include the trend
since, in general, we do not care the DGP of regressors and regressant in, for example, OLS. But, on the other hand, I am afraid
that we can not obtain the true value of b in the regression without the trend term since the estimator of b contains the trend,
so it can not be a consistent estimator.(I am afraid X'X dose not have probability limits)

(2) Although it is just my feeling, I think many studies did not include the trend in regression equations even if they showed
the significance of the trend term in the unit root tests, or they did not show any results on the trend term. How do you feel
this point?
(I am not convinced, actually...)

I am more than happy if you give me any suggestion.

Sincerely yours,

T_FIELD

[TomDoan]

Dear Dr. Tom Doan,

Thank you so much for your reply.

>you're correct that you have to include the trend in the regression in order for it to make sense as a cointegrating relation.

Can I ask some questions on your reply?

(1) I supposed x(t) and y(t) are stationary. (I supposed both variables reject DFtest).
Q1: Even so, do I have to care about cointegration? If so, could you tell me the reason or any (easy) text book?
Q2: When x(t) and y(t) are stationary, dose your answer change?

Cointegration is a property of integrated series, so no, if the series are stationary you don't have to worry about cointegration. However, if X is trend stationary and Y has no trend, then you will get exactly the same kind of result if you (mistakenly) run the regression without the trend---the coefficient on X will (in large samples) be effectively zero. The unit root process is between a non-trending stationary series and a trend stationary series in terms of the rate of increase of its cross product matrix. The only book that I know that goes through this in any detail is Hamilton's Time Series Analysis.

(What I want to know is, of course, the true value of parameter b. So, on the one hand, I think we need not include the trend
since, in general, we do not care the DGP of regressors and regressant in, for example, OLS. But, on the other hand, I am afraid
that we can not obtain the true value of b in the regression without the trend term since the estimator of b contains the trend,
so it can not be a consistent estimator.(I am afraid X'X dose not have probability limits)

What do you mean by the true value of b? You're focusing on the I(x)ness of the series, not on the economics of the model. If you have series that are even close to being non-stationary, then a simple y on x regression is almost certainly missing most of the dynamics of the data.

(2) Although it is just my feeling, I think many studies did not include the trend in regression equations even if they showed
the significance of the trend term in the unit root tests, or they did not show any results on the trend term. How do you feel
this point?
(I am not convinced, actually...)

That sounds like that would be wrong. Of course, if the model is estimated in differences, the trend gets downgraded to a constant so perhaps that's what you saw.

[T_FIELD]

Dear Dr. Tom Doan,

Thank you for your reply.

>What do you mean by the true value of b?

I am sorry to explain insufficiently.
I mentioned this b in my previous post.

>When we regress y(t) on X(t), is it correct to estimate y(t)=a+b*X(t)+u(t)?
>I worry about that y(t)=a+b*X(t)+c*trend+u(t) is correct, since the DGP of x(t)
> contains the trend term.

So, What I want to confirm is:
I have to estimate y(t)=a+b*X(t)+c*trend+u(t) when we regress y(t) on X(t), where,
the DGP of y(t) = a*y(t-1)+v(t), a<1 and v(t)~iid,
the DGP of x(t) = c*x(t-1)+w(t), c<1 and w(t)~iid.

Is this correct?

Thanking in advance for your trouble.

T_FIELD

[TomDoan]

I think you're missing the point. You can run any regression you want---the question is what is it that you are estimating? If you run OLS on

c=a+by+u

where c is consumption and y is income, is b the marginal propensity to consume? According to the textbook description of simultaneity, no. You can't make that type of structural interpretation of the coefficient because of the simultaneity in c and y. The OLS estimate of b will gives you a consistent estimate of...the coefficient in an OLS regression of c on y, which has no particular economic meaning.

Now, the theory of cointegration later showed that you could get a consistent estimate of the MPC by OLS if c and y are cointegrated, though consistent doesn't necessarily mean "good" in finite samples---you will likely get a much better estimate if you do something to deal with the omitted dynamics.

You're running a static regression on data which obviously has substantial omitted dynamics. I'm suggesting that you start thinking less about the statistical properties of the data, and start thinking more about the economics behind what you are doing.

[T_FIELD]

Thank you for your reply.

(1)
I am so sorry, but I mistook to write my question.

What I want to confirm is:
I have to estimate y(t)=a+b*X(t)+c*trend+u(t) when we regress y(t) on X(t), where,
the DGP of y(t) = a*y(t-1)+v(t), a<1 and v(t)~iid,
the DGP of x(t) = c*x(t-1)+d*trend+w(t), c<1 and w(t)~iid.

Dose this change affect your answer?

(2)
Yes, I might be confused, but I think my question is quite common.
What I am doing now is:

1: I tryed to estimate y(t)=a+b*X(t)+u(t)
2: To avoid spurious regression, I performed DFtest.
3: As the result, I found:
the DGP of y(t) = a*y(t-1)+v(t), a<1 and v(t)~iid,
the DGP of x(t) = c*x(t-1)+d*trend+w(t), c<1 and w(t)~iid.
4: I started to bother with this problem..

As you know, i simply want to obtain the consistent estimate of b.

I deeply appreciate for your kind assistance and discussion.
I am looking forward to your comments.

T_FIELD

[TomDoan]

Thank you for your reply.

(1)
I am so sorry, but I mistook to write my question.

What I want to confirm is:
I have to estimate y(t)=a+b*X(t)+c*trend+u(t) when we regress y(t) on X(t), where,
the DGP of y(t) = a*y(t-1)+v(t), a<1 and v(t)~iid,
the DGP of x(t) = c*x(t-1)+d*trend+w(t), c<1 and w(t)~iid.

Dose this change affect your answer?

(2)
Yes, I might be confused, but I think my question is quite common.
What I am doing now is:

1: I tryed to estimate y(t)=a+b*X(t)+u(t)
2: To avoid spurious regression, I performed DFtest.
3: As the result, I found:
the DGP of y(t) = a*y(t-1)+v(t), a<1 and v(t)~iid,
the DGP of x(t) = c*x(t-1)+d*trend+w(t), c<1 and w(t)~iid.
4: I started to bother with this problem..

As you know, i simply want to obtain the consistent estimate of b.

I deeply appreciate for your kind assistance and discussion.
I am looking forward to your comments.

T_FIELD

You started out with y(t)=a+b*X(t)+u(t). You then added a trend to the regression, not because y has a trend, or u has a trend, but because x has a trend. In the new form of the regression the b coefficient has a different meaning entirely---it's now not the effect of x on y, but of detrended x on y. If that's what you want, then you probably should have started with that. The regression you plan to run does give a consistent estimate of the coefficient on detrended x, though most people would simply detrend the x variable first; by throwing the trend on the right side, you are also "de-trending" y even though you've decided it has no trend.

[T_FIELD]

Thank you so much for your reply.
Now, I am convinced on what I was confused.

Actually, in my study, x is a policy variable, so, I feel it odd that x has a trend.
x has a trend in statistics sense, but x can not have a trend in economics sense.
This made me confused about interpretation of the estimated coefficient.
But I was now convinced that this discrepancy would be the difficult point of empirical study of economics.

I deeply and honestly appreciate for your kind assistance and discussion.

Yours Sincerely,

T_FIELD