# Constant term in time series econometric models built on 1-st differences

Both dependent and independent variables I deal with are nonstationary series that become stationary after differentiating them once.

The problem is that I assume that the dependent variable has a certain constant value which does not depend on the explanatory variables' changes and should be estimated as the model's constant term. But the constant term is lost in the process of differentiation.

Does anyone know any method of dealing with that kind of problem?

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As @IrishStat said it depends on the model. One way of recovering constant is to use the mean value of the residuals. Note that this method relies strongly on certain assumptions. Here is the illustration. Suppose your model is

$$Y_t=\alpha + \beta X_t + \varepsilon_t$$

with

$$E(\varepsilon_t|X_t)=0$$

and you estimate it by

$$\Delta Y_t=\beta \Delta X_t+\Delta \varepsilon_t.$$

Suppose your estimate $\hat\beta$ is unbiased (or at least consistent). Define

$$\hat{e}_t=Y_t-\hat\beta X_t.$$

Substituting the true model we have

$$\hat{e}_t=\alpha+\varepsilon_t+(\beta-\hat\beta)X_t,$$

hence

$$E(\hat{e}_t|X_t)=\alpha,$$

if $\hat\beta$ is unbiased or

$$\frac{1}{T}\sum_{t=1}^T\hat{e}_t\to \alpha,$$

if $\hat\beta$ is consistent.

So the natural estimate for $\alpha$ is

$$\hat{\alpha}=\frac{1}{T}\sum_{t=1}^T(Y_t-\hat\beta X_t).$$

Note that the model assumption is critical here. However with care this trick can be applied in general.

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There's some serial correlation and heteroskedasticity in the model which I handle using GLS but I guess heteroskedasticity wouldn't be the problem in using the trick you presented (thanks), yet serial correlation would, as the expected value of error term is the function of explanatory variables. Is this correct? –  robo Jun 2 '11 at 14:51
GLS doesn't render the error process i.i.d. , You need to incorporate ARIMA structure in conjunction with GLS to guarantee constant error variance in the presence of uncorrelated errors. Note well that neither of these transformations deal with Pulses,Level Shifts,Seasonal Pulses and/or Local Time trends that may remain in the residuals. Finally none of the above guarantees that the parameters have not changed over time. To review these concepts you might read some of my older responses to a myriad of previous questions. –  IrishStat Jun 2 '11 at 15:27
This will be probably a naive question, but wouldn't incorporation of AR rerm - a lagged explained variable into the model make the situation worse? Wouldn't it result in a biased estimator? And why using GLS/WLS does not eradicate heteroskedasticity? I mean,it's not intuitive for me - for observations with lower weights shouldn't we have lower error terms and the other way round? –  robo Jun 9 '11 at 12:25