Dear Colleagues,
For several years, I have been thinking about cointegration regression involving stationary variables as explanatory variables.
I am looking for comments on whether the following procedure is appropriate. I would be very grateful if you could advise me.
Setting:
- y(t)=a + b1*x1(t) + b2*x2(t) + b3*x3(t) +u(t).
- x1(t) and x2(t) are non-stationary, and x3(t) is stationary.
- Assume the cointegration relation y(t) = a + b1*x1(t) + b2*x2(t).
- x1(t) and x2(t) are not cointegrated.
I intend to perform the estimation and test using the following procedure.
Step 1: Estimate y(t)=a + b1*x1(t) + b2*x2(t) + z(t) by FMOLS.
b3*x3(t) will be included in z(t). The endogeneity problem may happen by the inclusion of x3(t) in z(t), but, FMOLS should control this problem.
Step 2: We follow Park-Phillips (1989).
Separately estimate y(t) = a + b1*x1(t) + b2*x2(t) + e(t) by OLS and get the (super)consistent estimator of a, b1 and b2 (a', b1' and b2'). Next, using these estimates, compute
y'(t) = y(t) - a' + b1'*x1(t) + b2'*x2(t)
Finally, we perform by IV,
y'(t) = c + b3*x3(t) + error
Note that a few minor assumptions are needed, See Park-Phillips (1989).
A possible problem is that x3(t) may not have zero-mean, because x3(t) is included in z(t). For now, I have no idea on it.
I would be happy if anybody gives me comments.
For several years, I have been thinking about cointegration regression involving stationary variables as explanatory variables.
I am looking for comments on whether the following procedure is appropriate. I would be very grateful if you could advise me.
Setting:
- y(t)=a + b1*x1(t) + b2*x2(t) + b3*x3(t) +u(t).
- x1(t) and x2(t) are non-stationary, and x3(t) is stationary.
- Assume the cointegration relation y(t) = a + b1*x1(t) + b2*x2(t).
- x1(t) and x2(t) are not cointegrated.
I intend to perform the estimation and test using the following procedure.
Step 1: Estimate y(t)=a + b1*x1(t) + b2*x2(t) + z(t) by FMOLS.
b3*x3(t) will be included in z(t). The endogeneity problem may happen by the inclusion of x3(t) in z(t), but, FMOLS should control this problem.
Step 2: We follow Park-Phillips (1989).
Separately estimate y(t) = a + b1*x1(t) + b2*x2(t) + e(t) by OLS and get the (super)consistent estimator of a, b1 and b2 (a', b1' and b2'). Next, using these estimates, compute
y'(t) = y(t) - a' + b1'*x1(t) + b2'*x2(t)
Finally, we perform by IV,
y'(t) = c + b3*x3(t) + error
Note that a few minor assumptions are needed, See Park-Phillips (1989).
A possible problem is that x3(t) may not have zero-mean, because x3(t) is included in z(t). For now, I have no idea on it.
I would be happy if anybody gives me comments.
