# What does this notation mean?

Could anybody tell me what is this model? What does it mean?

$$\Delta Y = C + a(Y_{t-1} + bX_{t-1}) + c\Delta Y_{t-1} + d\Delta X_{t-1} + \varepsilon_t$$

Where $\Delta$ is the differencing operator, $a, b, c,$ and $d$ are coefficients, and $t$ is time.

• I do not quite get the question, but I will try. Although a pair of parentheses is missing, this is one equation from a vector error correction model (VECM, or VEC model). Did I answer your question? – Richard Hardy Aug 18 '15 at 18:49
• Thank you. I am learning cointegration test. I used EG test the regression function is Y = c + aX+e, e is error term. but recently someone suggests me another model : △Y= C + aY(t-1) + bX(t-1) + c△Y(t-1）+ d△Y(t-2) + e△Y（t-3)...... + f△X（t-1) + g△X(t-2) + h △X（t-3)......The first step of EG test is building a regression function, can I use this model and test the stationary of residual? – Jasmine Aug 18 '15 at 19:19
• For the first step of the Engle-Granger cointegration testing, use the model $Y_t=\beta_0+\beta_1 X_t+\varepsilon_t$ and explore the stationarity (or lack thereof) of the residual. Meanwhile, the VECM equation you have up there should not be used for this purpose. But once you establish presence of cointegration, you can specify a VECM and proceed with it. – Richard Hardy Aug 18 '15 at 20:05

Technically the answer by @Aksakal is correct, but I suspect a different answer may be more useful given the context. Since cointegration was mentioned and the original model representation resembles very closely an equation from a vector error correction model (VECM) which is used for modelling cointegrated processes, I would guess the model in mind actually is VECM.

A full bivariate VECM of autoregressive order $p$ would look as follows:

\begin{align} \Delta Y_t &= \alpha_1(Y_{t-1}+\beta X_{t-1}) + \gamma_{1,11} \Delta Y_{t-1} + \gamma_{1,12} \Delta X_{t-1} + \dotsb + \\ &\quad\ \gamma_{p,11} \Delta Y_{t-p} + \gamma_{p,12} \Delta X_{t-p} + \varepsilon_{1,t} \\[10pt] \Delta X_t &= \alpha_2(Y_{t-1}+\beta X_{t-1}) + \gamma_{2,21} \Delta Y_{t-1} + \gamma_{2,22} \Delta X_{t-1} + \dotsb + \\ &\quad\ \gamma_{p,21} \Delta Y_{t-p} + \gamma_{p,22} \Delta X_{t-p} + \varepsilon_{2,t} \end{align}

The term $Y_{t-1}+\beta X_{t-1}$ is known as the error correction term, and vector $(1,\beta)$ is the cointegrating vector. The coefficients $\alpha_1$, $\alpha_2$ are known as loading factors. This model shows how a pair of cointegrated variables ($Y_t$,$X_t$) develop over time. The error correction term multiplied by the loading terms acts to hold $Y_t$ close to $X_t$, and vice versa (if the signs of $\alpha$s and $\beta$ are as expected).

I am not going to expand on the mechanics of the model further, but given the keywords you should be able to find relevant literature.

Yes, delta signs are differences, and time is $t$. Plugging the missing notation this is what you get:

\begin{align} Y_{t}-Y_{t-1} &= C + aY_{t-1} + bX_{t-1} + c( Y_{t-1}-Y_{t-2}) + d( X_{t-1}-X_{t-2}) \\ Y_{t} &= C + (1+a+c)Y_{t-1} + (b+d)X_{t-1} -cY_{t-2} -dX_{t-2} \end{align}

Which is ARX(2) autoregressive order 2 process with exogenous covariates: $$Y_t = C + \phi_1 Y_{t-1} +\phi_2 Y_{t-2} + \beta_0 X_{t-1} + \beta_2 X_{t-2} + \varepsilon_t$$

Notice the lags on the exogenous variable ($X_{t-1},\ X_{t-2}$). They probably come with negative coefficients, which will make this model look like some kind of error correction model, as Richard Hardy suggested