<p>Let's explore this with a simple example.  In the table below, the first 4 columns calculate $Y$ from $P$ using the model:</p>

<p>$$Y = \beta_0 + \beta_1P$$</p>

<p>with $\beta_0=5$ and $\beta_1=2$ (in the interests of simplicity there is no $P^2$ term).  </p>

<p><a href="https://i.sstatic.net/LKPpJ.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LKPpJ.jpg" alt="enter image description here"></a></p>

<p>Because of the way the $Y$-values have been calculated, $Y$ and $P$ are perfectly correlated, and a regression of $Y$ on a constant and $P$ estimates $\beta_1$ as exactly 2, with standard error zero.</p>

<p>Now we introduce greater realism by adding some random disturbances.  The column headed $RND$ contains random numbers within the range $(-10,10)$, and $YRAND = Y + RND$. A regression of $YRAND$ on a constant and $P$ estimates $\beta_1$ as 1.85 with a standard error of 0.46 and p-value 0.004, ie significantly different from zero at even the 1% level.</p>

<p>Suppose now that we focus on the changes in variables shown in the last 3 columns of the above table.  A regression of $\Delta Y$ on $\Delta P$ with, as Alecos Papadopoulos explains in his answer, no constant term, again estimates $\beta_1$ as exactly 2, with standard error zero. </p>

<p>So will the regression of $\Delta YRAND$ on $\Delta P$ yield the same results for $\beta_1$ as that of $YRAND$ on $P$?  No, because the effect of the random disturbances on changes in $Y$ is much greater, proportionally, than their effect on the absolute values of $Y$.  In fact, this regression estimates $\beta_1$ as 3.12, with a standard error of 2.31 and a p-value of 0.21, ie not significantly different from zero at the 5% significance level.</p>

<p>So a likely interpretation of the results is simply that this is normal behaviour when there is a degree of random variation in the dependent variable.  </p>

CC BY-SA 3.0
2017-08-04T21:59:28.703Z
This explanation is excellent. Really like how you simulated a data set also. Thank you!
EconJohn
CC BY-SA 3.0
2017-08-04T21:42:29.347Z
2017-08-04T21:42:29.347Z
Adam Bailey
<p>If the levels specification is deemed acceptable,</p>

<p>$$y_t = \beta_0 + \beta_1 x_t + \beta_2 x_t^2$$</p>

<p>then it follows that the first-difference specification <em>should not include a constant term</em> in order to be methodologically consistent,</p>

<p>$$\Delta y_t = \beta_1\Delta x_t + \beta_2\Delta x_t^2$$</p>

<p>since $\Delta \beta_0 = \beta_0 -\beta_0 = 0$.</p>

<p>I suggest you run the specification of first-differences without intercept and see what happens.</p>

<p>Note: the slope coefficient estimates you should obtain in the first-difference specification without intercept should be close to the corresponding estimates you obtained in the levels-estimation. Otherwise, the maintained hypothesis of constant slope coefficients becomes questionable, or the model has other misspecification issues. </p>

<p>(this is not to be confused with the case where we run a regression without constant term but with the dependent variable and the regressors centered on their sample means -there we would obtain the exact same slope estimates with the levels specification that includes a constant term, as a matter of algebraic property of least-squares estimation).</p>

CC BY-SA 3.0
2017-08-04T15:10:55.863Z
Is there an interpretation of such a regression?
EconJohn
CC BY-SA 3.0
2017-08-04T17:14:27.980Z
@EconJohn The first-difference you mean?
Alecos Papadopoulos
CC BY-SA 3.0
2017-08-04T18:57:34.170Z
My bad I just looked up the terminology. I didn't know i was getting my feet wet in panel data methods. Thank you!
EconJohn
CC BY-SA 3.0
2017-08-05T12:42:09.103Z
@EconJohn, first-difference is not panel data term (AFAIK), it is a time series term for transforming $x_t$ into $\Delta x_t:=x_t-x_{t-1}$.
Richard Hardy
CC BY-SA 3.0
2017-08-03T20:40:06.130Z
2017-08-05T15:38:45.463Z
2017-08-05T15:38:45.463Z
Alecos Papadopoulos
<p>I'm experimenting with some data in R and I've found that though there is statistical significance between two variables, however their changes are not statistically significant. </p>

<p>I first ran a standard regression of revenue on price, adding a quadratic term to account for diminishing returns from increase in price. Giving us the formula:</p>

<p>$$y_{Revenue}=\beta_0+\beta_1Price+\beta_2Price^2$$</p>

<p>The results which were produced are:</p>

<pre><code>&gt; summary(lm(Wage~Price+I(Price^2)))

Call:
lm(formula = Wage ~ Price+ I(Price^2))

Residuals:
Min      1Q  Median      3Q     Max 
-131.87  -87.77  -27.60   44.15  244.66 

Coefficients:
           Estimate Std. Error t value Pr(&gt;|t|)    
(Intercept)  -1.650e+03  2.645e+02  -6.238 5.44e-06 ***
Price         3.640e-01  3.640e-02   9.999 5.28e-09 ***
I(Price^2)   -1.026e-05  1.129e-06  -9.086 2.41e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 116.9 on 19 degrees of freedom
  (7 observations deleted due to missingness)
Multiple R-squared:  0.8816,    Adjusted R-squared:  0.8691 
F-statistic: 70.72 on 2 and 19 DF,  p-value: 1.577e-09
</code></pre>

<p>The second regression I ran was a regression of the change in revenue on change in price. Giving the formula:
$$\Delta y_{Revenue}=\alpha_0+\alpha_1 \Delta Price+\alpha_2 \Delta Price^2$$</p>

<pre><code>Call:
lm(formula = diff(Revenue) ~ diff(Price) + diff(I(Price^2)))

Residuals:
   Min     1Q Median     3Q    Max 
-82.52 -42.55 -11.98  19.20 142.36 

Coefficients:
                     Estimate Std. Error t value Pr(&gt;|t|)   
(Intercept)         5.093e+01  2.649e+01   1.923  0.07046 . 
diff(Price)         1.343e-01  7.165e-02   1.874  0.07727 . 
diff(I(Price^2))   -4.987e-06  1.691e-06  -2.950  0.00857 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 62.29 on 18 degrees of freedom
  (7 observations deleted due to missingness)
Multiple R-squared:  0.4521,    Adjusted R-squared:  0.3912 
F-statistic: 7.426 on 2 and 18 DF,  p-value: 0.004449
</code></pre>

<p>Why do these variables lose their degree of  statistical significance while in the regular regression they are significant on the 1% level and how do you interpret such results economically?</p>

CC BY-SA 3.0
2017-08-03T19:53:07.803Z
Spurious regression (relationship, correlation)?
Richard Hardy
CC BY-SA 3.0
2017-08-03T20:17:57.033Z
@RichardHardy Honestly just messing around with the data. Im interested as to why change in these variables aren't as statistically significant when their actual state is significant on the 1% level.
EconJohn
CC BY-SA 3.0
2017-08-03T20:18:15.797Z
@RichardHardy so you think it means nothing?
EconJohn
CC BY-SA 3.0
2017-08-04T05:19:51.643Z
This was just an idea what could be going on. I cannot see whether your time series are integrated, but if they are, then this could be the case.
Richard Hardy
<applied-econometrics>
Interpretation of a differenced regression
EconJohn
# Interpretation of a differenced regression

(Asked by: username_0 on 2017-08-03)

I'm experimenting with some data in R and I've found that though there is statistical significance between two variables, however their changes are not statistically significant. 
I first ran a standard regression of revenue on price, adding a quadratic term to account for diminishing returns from increase in price. Giving us the formula:
$$y_{Revenue}=\beta_0+\beta_1Price+\beta_2Price^2$$
The results which were produced are:
> summary(lm(Wage~Price+I(Price^2)))

Call:
lm(formula = Wage ~ Price+ I(Price^2))

Residuals:
Min      1Q  Median      3Q     Max 
-131.87  -87.77  -27.60   44.15  244.66 

Coefficients:
           Estimate Std. Error t value Pr(>|t|)    
(Intercept)  -1.650e+03  2.645e+02  -6.238 5.44e-06 ***
Price         3.640e-01  3.640e-02   9.999 5.28e-09 ***
I(Price^2)   -1.026e-05  1.129e-06  -9.086 2.41e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 116.9 on 19 degrees of freedom
  (7 observations deleted due to missingness)
Multiple R-squared:  0.8816,    Adjusted R-squared:  0.8691 
F-statistic: 70.72 on 2 and 19 DF,  p-value: 1.577e-09

The second regression I ran was a regression of the change in revenue on change in price. Giving the formula:
$$\Delta y_{Revenue}=\alpha_0+\alpha_1 \Delta Price+\alpha_2 \Delta Price^2$$
Call:
lm(formula = diff(Revenue) ~ diff(Price) + diff(I(Price^2)))

Residuals:
   Min     1Q Median     3Q    Max 
-82.52 -42.55 -11.98  19.20 142.36 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)   
(Intercept)         5.093e+01  2.649e+01   1.923  0.07046 . 
diff(Price)         1.343e-01  7.165e-02   1.874  0.07727 . 
diff(I(Price^2))   -4.987e-06  1.691e-06  -2.950  0.00857 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 62.29 on 18 degrees of freedom
  (7 observations deleted due to missingness)
Multiple R-squared:  0.4521,    Adjusted R-squared:  0.3912 
F-statistic: 7.426 on 2 and 18 DF,  p-value: 0.004449

Why do these variables lose their degree of  statistical significance while in the regular regression they are significant on the 1% level and how do you interpret such results economically?

**Tags:** applied-econometrics

> **Comments:**
> **username_1**: Spurious regression (relationship, correlation)?  
> **username_0**: @RichardHardy Honestly just messing around with the data. Im interested as to why change in these variables aren't as statistically significant when their actual state is significant on the 1% level.  
> **username_0**: @RichardHardy so you think it means nothing?  
> **username_1**: This was just an idea what could be going on. I cannot see whether your time series are integrated, but if they are, then this could be the case.  

---

## 2 answers

### Answer by username_3 on 2017-08-03. Score: 3 | Accepted answer

If the levels specification is deemed acceptable,

$$y\_t = \beta\_0 + \beta\_1 x\_t + \beta\_2 x\_t^2$$

then it follows that the first-difference specification *should not include a constant term* in order to be methodologically consistent,

$$\Delta y\_t = \beta\_1\Delta x\_t + \beta\_2\Delta x\_t^2$$

since $\Delta \beta\_0 = \beta\_0 -\beta\_0 = 0$.

I suggest you run the specification of first-differences without intercept and see what happens.

Note: the slope coefficient estimates you should obtain in the first-difference specification without intercept should be close to the corresponding estimates you obtained in the levels-estimation. Otherwise, the maintained hypothesis of constant slope coefficients becomes questionable, or the model has other misspecification issues.

(this is not to be confused with the case where we run a regression without constant term but with the dependent variable and the regressors centered on their sample means -there we would obtain the exact same slope estimates with the levels specification that includes a constant term, as a matter of algebraic property of least-squares estimation).

> **Comments:**
> **username_0**: Is there an interpretation of such a regression?  
> **username_3**: @username_0 The first-difference you mean?  
> **username_0**: My bad I just looked up the terminology. I didn't know i was getting my feet wet in panel data methods. Thank you!  
> **username_1**: @username_0, first-difference is not panel data term (AFAIK), it is a time series term for transforming $x_t$ into $\Delta x_t:=x_t-x_{t-1}$.  

---

### Answer by username_2 on 2017-08-04. Score: 2

Let's explore this with a simple example. In the table below, the first 4 columns calculate $Y$ from $P$ using the model:

$$Y = \beta\_0 + \beta\_1P$$

with $\beta\_0=5$ and $\beta\_1=2$ (in the interests of simplicity there is no $P^2$ term).

[![enter image description here](https://i.sstatic.net/LKPpJ.jpg)](https://i.sstatic.net/LKPpJ.jpg)

Because of the way the $Y$-values have been calculated, $Y$ and $P$ are perfectly correlated, and a regression of $Y$ on a constant and $P$ estimates $\beta\_1$ as exactly 2, with standard error zero.

Now we introduce greater realism by adding some random disturbances. The column headed $RND$ contains random numbers within the range $(-10,10)$, and $YRAND = Y + RND$. A regression of $YRAND$ on a constant and $P$ estimates $\beta\_1$ as 1.85 with a standard error of 0.46 and p-value 0.004, ie significantly different from zero at even the 1% level.

Suppose now that we focus on the changes in variables shown in the last 3 columns of the above table. A regression of $\Delta Y$ on $\Delta P$ with, as username_3 explains in his answer, no constant term, again estimates $\beta\_1$ as exactly 2, with standard error zero.

So will the regression of $\Delta YRAND$ on $\Delta P$ yield the same results for $\beta\_1$ as that of $YRAND$ on $P$? No, because the effect of the random disturbances on changes in $Y$ is much greater, proportionally, than their effect on the absolute values of $Y$. In fact, this regression estimates $\beta\_1$ as 3.12, with a standard error of 2.31 and a p-value of 0.21, ie not significantly different from zero at the 5% significance level.

So a likely interpretation of the results is simply that this is normal behaviour when there is a degree of random variation in the dependent variable.

> **Comments:**
> **username_0**: This explanation is excellent. Really like how you simulated a data set also. Thank you!  

---

