# ARDL/Error Correction Model: long vs. short run, restricted vs. unrestricted

I have a few questions about unrestricted error correction models.

The UECM for a model where $Y$ is the dependent variable and $x$ is the sole independent variable is given by $$\Delta Y_{t}=\alpha_{0}+\sum_{n=1}^{N}\beta_1\Delta y_{t-i}+\sum_{n=1}^{N}\beta_2\Delta x_{t-i}+\gamma_{1}y_{t-1}+\gamma_{2}x_{t-1}.$$ Here are my questions:

1. Why are the $\gamma$s considered long run coefficients and the $\beta$s considered the short run coefficients? What is the idea behind interpreting coefficients on level variables as long run and those on differenced variables as short run?

2. How would I interpret the short run and log run coefficients? For instance, the $\beta$ coefficients on a cross sectional OLS provide information on how a unit change in the independent variable changes the dependent variable. Would I be interpreting the long run and short run coefficients of the UECM in the same way? If so, what does the long and short term have anything to do with this interpretation? And what span of time is considered long vs. short term in the context of these model?

3. Why is this model called an unrestricted model? What about it makes it different from a regular error correction model?

## 3 Answers

1. In the long run the first differences are taken as zero and the long-run equation reduces to

$\gamma_1y+\gamma_2x$=0 which is the long run relationship between the variables. The $\gamma$'s define this long run relationship. The $\beta$'determine the short run adjustment to this equilibrium.

2.See 1. The $\beta$'s are a measure of the persistence of the variable. I don't think that you need to pay particular attention to individual values. In the context of the model the long run relationship can be interpreted as your panel equation. There is no set rule determining the short and long run. One can estimate the half life of a disturbance to equilibrium from the estimated coefficients. This will be different for every model

3 I am not sure that I understand this question. Are you differentiating between a model where the constant is constrained to the ecm or where the constant is unrestricted?

• John, that was very helpful. I have one clarification question. Why are the first differences set to zero in the long run? Feb 16, 2015 at 15:35
• The long run is the equilibrium or steady state of the system arrived at when things stop changing. In such cases all first differences are zero. This is the most common case encountered in macro-econometrics. If the system had an equilibrium with a variable growing at a constant rate then the first difference of the growth rate is zero at equilibrium. Feb 17, 2015 at 20:09

On question 1:

In this setting we assume that $$Y$$ and $$x$$ are not stationary. If they were stationary there'd be little sense in talking about a long-run relationship -- $$E[Y_t]$$ would be some $$\mu$$ for all $$t$$.

Instead, what is assumed (and should be tested) is that $$Y$$ and $$x$$ cointegrate, i.e. that there's some linear combination $$\gamma_1 Y + \gamma_2 x + \gamma_3 \cdot 1$$ (these models are always affine, but linear if we assume a constant column) that is stationary. That's your long-term relationship.

Then something caled the Granger Representation Theorem guarantees that a cointegration relationship can be estimated with an Error Correction Model.

1. It's called unrestricted because all the long run relationship'varaiables are specified. So, there is no restriction about the presence of any variables. So, when you replace this long run relationship by its Residuals, the model becomes ECM because the ECT (Error Correction Term) corrects the disequilibrum happened in a short period, by bringing the situation to a steady statement.