<p>Consider the two following models</p>
<p><span class="math-container">$$ \Delta y = \beta_0 + \beta_1 y_1 + \beta_2 \Delta x^1 + \beta_3 \Delta x^2 + ...  $$</span></p>
<p>and</p>
<p><span class="math-container">$$ \Delta y = \beta_0 + \beta_1 \Delta x^1 + \beta_2 \Delta x^2 + ...  $$</span></p>
<p>where <span class="math-container">$\Delta y = y_2 - y_1$</span> and <span class="math-container">$\Delta x^i = x^i_2 - x^i_1$</span>. Let's say I want to understand the effects of changes in <span class="math-container">$x$</span> on changes in <span class="math-container">$y$</span> and am doing a simple OLS. Under what circumstances, is it preferable to choose the first model over the second and vice versa?</p>
<p>I cannot seem to find an exact treatment of the problem anywhere. Any hints/ideas are also welcome. Thanks.</p>

CC BY-SA 4.0
2024-07-07T04:56:20.930Z
Hi: I'm not experience with first differenced regressions but I see the idea. My question is, if it's really the case that you are interested in how , $x_2-x_1$, ( well more than one $x$ but for simplicity, I'll just think of it as $x_2 - x_1$) is related to $y_2 - y_1$, then why should the initial level of $y_1$ matter ? In fact, what you are really doing in (1) is $y_2 - (1-\beta) y_1 = \beta_0 + \beta_2 \Delta x^{1} + \ldots$. But I don't think you would want that because you are then regressing first differences on what is no longer first differences.
mark leeds
CC BY-SA 4.0
2024-07-07T05:02:22.520Z
Also, first differences regression seems to require that one choose what is the dependent variable and what is the independent variable. I would do it both ways because you could get inconsistent results. There's a similar problem with the Engle-Granger cointegration framework in terms of which to choose as which.
mark leeds
CC BY-SA 4.0
2024-07-08T17:22:49.450Z
Hi. Thanks for the comments. I might be mistaken, but the reason I include initial $y$ is because I thought it would control for initial conditions and unobserved heterogeneity among the data points. When I run both models, I find that the coefficient on $y_1$ is significant, suggesting there is some effect. The $R^2$ is also improved when I include $y_1$. I guess I am not sure if I can rely on the coefficients of the first model, but the coefficient of interest is quite similar across models, although some other coefficients are quite different.
Mwazr A
CC BY-SA 4.0
2024-07-09T02:50:00.790Z
The problem I see is that including it makes it no longer a first differences regression. If there's observed heterogeneity in the response, maybe you want to leave it out $\beta y_1$ but use GLS to estimate the model ? Like I said, I have no experience with first differences regression but I don't think including the initial level of $y$ is the way to go because it changes the underlying theme of the model which is first differences.
mark leeds
CC BY-SA 4.0
2024-07-09T23:43:49.030Z
Yeah I agree. I see that there is a negative coefficient on $y_1$ that is very significant showing that there is something going on there. I will try GLS but maybe some non linear regression could also be useful. Need to think about this a little more.  

Thanks for your comments.
Mwazr A
CC BY-SA 4.0
2024-07-11T03:29:57.487Z
Hi: if $\beta$ is negative and not near zero, then it could be that the differences of $x$ have an effect on $(y_2 - (1-\beta y_1))$ which isn't out of the question. So, I agree that there might be something interesting going on there. As the model stands, it's linear so don't introduce non-linear regression unless you're looking to build a non-linear model. Good luck.
mark leeds
<econometrics><applied-econometrics>
What are the Advantages and Disadvantages of including the initial level of the dependent variable in a first difference regression?
Mwazr A
# What are the Advantages and Disadvantages of including the initial level of the dependent variable in a first difference regression?

(Asked by: username_0 on 2024-07-07)

Consider the two following models

$$ \Delta y = \beta\_0 + \beta\_1 y\_1 + \beta\_2 \Delta x^1 + \beta\_3 \Delta x^2 + ... $$

and

$$ \Delta y = \beta\_0 + \beta\_1 \Delta x^1 + \beta\_2 \Delta x^2 + ... $$

where $\Delta y = y\_2 - y\_1$ and $\Delta x^i = x^i\_2 - x^i\_1$. Let's say I want to understand the effects of changes in $x$ on changes in $y$ and am doing a simple OLS. Under what circumstances, is it preferable to choose the first model over the second and vice versa?

I cannot seem to find an exact treatment of the problem anywhere. Any hints/ideas are also welcome. Thanks.

**Tags:** econometrics, applied-econometrics

> **Comments:**
> **username_1**: Hi: I'm not experience with first differenced regressions but I see the idea. My question is, if it's really the case that you are interested in how , $x_2-x_1$, ( well more than one $x$ but for simplicity, I'll just think of it as $x_2 - x_1$) is related to $y_2 - y_1$, then why should the initial level of $y_1$ matter ? In fact, what you are really doing in (1) is $y_2 - (1-\beta) y_1 = \beta_0 + \beta_2 \Delta x^{1} + \ldots$. But I don't think you would want that because you are then regressing first differences on what is no longer first differences.  
> **username_1**: Also, first differences regression seems to require that one choose what is the dependent variable and what is the independent variable. I would do it both ways because you could get inconsistent results. There's a similar problem with the Engle-Granger cointegration framework in terms of which to choose as which.  
> **username_0**: Hi. Thanks for the comments. I might be mistaken, but the reason I include initial $y$ is because I thought it would control for initial conditions and unobserved heterogeneity among the data points. When I run both models, I find that the coefficient on $y_1$ is significant, suggesting there is some effect. The $R^2$ is also improved when I include $y_1$. I guess I am not sure if I can rely on the coefficients of the first model, but the coefficient of interest is quite similar across models, although some other coefficients are quite different.  
> **username_1**: The problem I see is that including it makes it no longer a first differences regression. If there's observed heterogeneity in the response, maybe you want to leave it out $\beta y_1$ but use GLS to estimate the model ? Like I said, I have no experience with first differences regression but I don't think including the initial level of $y$ is the way to go because it changes the underlying theme of the model which is first differences.  
> **username_0**: Yeah I agree. I see that there is a negative coefficient on $y_1$ that is very significant showing that there is something going on there. I will try GLS but maybe some non linear regression could also be useful. Need to think about this a little more.    Thanks for your comments.  
> **username_1**: Hi: if $\beta$ is negative and not near zero, then it could be that the differences of $x$ have an effect on $(y_2 - (1-\beta y_1))$ which isn't out of the question. So, I agree that there might be something interesting going on there. As the model stands, it's linear so don't introduce non-linear regression unless you're looking to build a non-linear model. Good luck.  

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## 0 answers

