Help understanding how the cointegration equation for VECM models are derived

Question

I am learning about Vector Error Correction Models from Sean Becketti's "Introduction to Time Series using Stata". While I know how to run the Stata commands to estimate the VECM, I have no idea why the author is interpreting certain components of the VECM the way he is. There are no derivations of the model, and thus, it is very hard for me to accept his interpretation of the model.

The author starts by assuming that two cointegrated series obey the following relationship.

$$ y_{t} = \alpha_{1}y_{t-1}+\gamma_{0}z_{t}+\gamma_1z_{t-1}+\epsilon_{t} $$

He then goes on to subtract $y_{t-1}$ from both sides and adds and subtracts $z_{t-1}$ from the RHS. Rearrangement yields the following.

$$ \Delta y_{t}=(\alpha_{1}-1)y_{t-1}+\gamma_{0}\Delta z_{t}+(\gamma_{0} + \gamma_{1})z_{t-1}+\epsilon_{t} $$

Rearranging again gives,

$$ \Delta y_{t} = \gamma_{0}\Delta z_{t}-\underbrace{\lambda(y_{t-1}-\theta z_{t-1})}_{Cointegrating Equation}+\epsilon_{t} $$

Where $\lambda=(1-\alpha_{1})$ and $\theta = \frac{\gamma_{0}+\gamma_{1}}{\alpha_{1}-1}$

Here are my questions for you guys.

1) Why is $(y_{t-1}-\theta z_{t-1})$ the long run relationship? The author never explains this in the book.

2) The author says that non-zero values of the cointegrating eq should be interpreted as errors. Why is this?

3) Lastly, can you direct me to a book or paper that derives this model? I am looking for something that explains it at the level of detail that introductory Econometrics texts explain OLS models. So for instance, if I was learning the OLS, I would like a book to derive the SSE, take the derivative of the SSE, set it equal to 0 and solve for the betas, and then explain the intuition behind why those steps had been taken.

Thanks a ton. Look forward to hearing back from you.

Answer 1

I will provide a much-simplified example as a rough illustration if you don't mind lack of rigor. This is just one special case, but I hope it will be enlightening.

Let $z_t$ be a random walk, a cumulative sum of an independently and identically distributed (i.i.d.) series $\xi_t$: $z_t=\sum\limits_{i=1}^t \xi_t$.

Let $y_t$ be generated as follows: $y_t:=\theta z_t+u_t$, where $u_t$ is another i.i.d. series.

If you plot $y_t$ and $\theta z_t$ on one graph, you will see that they track each other closely up to an error (which is, of course, $u_t$). The pair $z_t$ and $y_t$ is (almost) the simplest example of a pair of cointegrated variables.

Once again: since you know how $y_t$ was generated, you will expect it to be close to $\theta z_t$ all the time. This will be clear in the long run; for just a few data points, it is hard to see that $y_t$ and $\theta z_t$ indeed always come back close to each other even if there are some occasional larger gaps between them. That's how this long term relationship comes about.

Calling the term $(y_t-\theta z_t)$ an error just means calling $u_t$ an error. If $y_t$ and $z_t$ have some subject-matter interpretations while $u_t$ does not, then it kind of justifies the name.

Now in reality you rarely know with certainty how $y_t$ was generated. However, if you plot $y_t$ and $z_t$ on the same graph and notice that they are roughly proportional, you might get curious. (You then can estimate the unknown $\theta$, subsequently obtain an estimate of $u_t$ and plot it to visually check whether $u_t$ looks like an i.i.d. series. And then do some formal testing to establish your finding more seriously.) Ideally, subject-matter knowledge will tell you whether you can expect $y_t$ and $z_t$ to be cointegrated so that you need not rely purely on statistics.

I hope that by now you have some clue of what is happening there with $z_t$ and $y_t$. I am not sure about the terminology Sean Becketti uses (the way you put it in your questions does not correspond exactly to what I am used to) but with some imagination you can accept it, I guess.

There are a few good posts about VECM here on Cross Validated, you might benefit by checking them out (using the vecm tag or searching for "VECM"). Also, I like the treatment of cointegration in Davidson & MacKinnon "Econometric Theory and Methods" and Zivot's lecture notes. Good luck!

Answer 2

Let me add to the great explanation by Richard. Or actually let me try to explain the basic concept in (even) simpler (and therefore may be a little bit less correct) terms, since you asked for a very simple explanation.

1. Let's start with the cointegration relationship:

   The two time series are cointegrated, which basically means that they both change over time ("they are non-stationary") but in a "similar" way. If they both "change in a similar way" this implies that the relationship between them does not change. Translating this into formulas: $y_t = θ z_t$ represents the relationship between the two time series. If $θ$ is the same for all t this means that the relationship does not change.

   However since we are dealing with real world data the time series the described relationship is only the long run "average" relationship. For any $z_t$ the actual value of $y_t$ will therefore either be a little bit above or below the value that we would have expected based on our equation. This is captured in the error term $u_t$ (which OLS tries to minimize): $y_t = θ z_t + u_t$

   Since we claimed that the relationship (between $y$ and $z$) does not change over time, $u_t$ has to be stationary.

   Therefore the long rund relationship/deviation from the long run relation ship at $t$ is represented by $y_{t−1}−\theta z_{t−1}=u_t$

2. The actual Error Correction Model:

   Since we said that relationship $y_{t−1}−\theta z_{t−1}=u_t$ is stationary this means that if we deviate from this relationship in one period, it is likely that we will correct this deviation over the following periods. So if as an example $y_t$ was 1 unit higher than expected by the long run relationship the error in $t=1$ would be $u_t=1$. We would then expect $\Delta y_{t+1}=y_{t+1}-y_t < 0$ so that the value of y would be corrected "downwards" after it was "too high" in the previous period.

   To achieve this effect the error ("deviation from long run relationship") of the previous period is included in the Error correction model. The coefficient $\lambda$ provides information on how quickly this deviation is "corrected": $\lambda(y_{t−1}−θz_{t−1}) = \lambda (u_{t-1})$

Hope this naive explanation helped to understand the basic concept to then be able to go on to Richard's answer to understand the concept correctly :)