<p>I came across the answer to the validity of univariate modelling in economics (albet indirectly) from the book called <a href="http://press.uchicago.edu/ucp/books/book/chicago/R/bo5947912.html" rel="nofollow noreferrer">A Rational Expectations Approach to Macroeconometrics</a>. This answer is based on the material in the book, for a better understanding I reccomend actually going through it.</p>

<blockquote>
  <p>In short the validity of ARIMA models is based on the assumption that the market is always in equilibrium or <strong>efficient</strong><sup>1</sup>.</p>
</blockquote>

<p>In order to understand how use of ARIMA models in Econometrics is a valid way to forecast economic variables, we must understand models which considers the case of for <strong>rational expectations</strong> (other wise referred to as <strong>efficient markets</strong>).</p>

<p>your standard efficient markets model is:</p>

<p>$$y_t=\tilde{y}_t+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$$
where:</p>

<ul>
<li>$y_t=$ Economic variable of interest at time $t$ like unemployment or real output.</li>
<li>$\tilde{y}_t=$Natural rate or equilibrium of economic variable at time $t$</li>
<li>$\beta_i=$ coefficents.</li>
<li>$X_{t-i}=$ an aggreagte demand variable like money growth, Inflation or Nominal GDP growth.</li>
<li>$X^e_{t-i}=$ Anticipated aggregate demand conditional on knowing all  information at point $t-i$</li>
<li>$\epsilon_t=$ an error term</li>
</ul>

<p>This model accounts for variability that can be present when a market does not have all information pertaining to $y_t$.</p>

<p>combining this formula using a specification used with the Lucas supply function:
$$\tilde{y}=\sum_{i=1}^N\lambda_iy_{t-i}$$</p>

<p>combining the two equations we get:</p>

<p>$$y_t=\sum_{i=1}^N\lambda_iy_{t-i}+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$$</p>

<p>This model tells us that if rational expectations hold,the relationship of  $y_t=\sum_{i=1}^N\lambda_iy_{t-i}$, thus providing validity for the use of uni variate models in economics.</p>

<p>For more reading on this topic, below is posted links to the three chapters which helped me:<br>
1)<a href="http://www.nber.org/chapters/c10241.pdf" rel="nofollow noreferrer">http://www.nber.org/chapters/c10241.pdf</a><br>
2)<a href="http://www.nber.org/chapters/c10243.pdf" rel="nofollow noreferrer">http://www.nber.org/chapters/c10243.pdf</a><br>
3)<a href="http://www.nber.org/chapters/c10244.pdf" rel="nofollow noreferrer">http://www.nber.org/chapters/c10244.pdf</a></p>

<p><strong>UPDATE:</strong> Upon Contacting <a href="https://twitter.com/MyProgDirector/status/971261126657888256" rel="nofollow noreferrer">Dr. Paul Middleditch on Twitter</a> he says that the RHS variables must be replaced with a proxy which is normally the value of the previous period. ARIMA methodology satisfies such constraints.<sup>2</sup></p>

<hr>

<p><sub>1. Mathematically this statement can be written as:
$$\mathbb{E}_m(X_t|\phi_{t-1})=\mathbb{E}(X_t|\phi_{t-1})$$</sub></p>

<p><sub> where the expected value given by the market of $X_t$ conditional on $\phi_{t-1}$ the set of all available information at time $t-1$, is equal to the true expected value of $X_t$</sub></p>

<p><sub>It follows logically that $\mathbb{E}[X_t-\mathbb{E}_m(X_t)|\phi_{t-1}]=0$ </sub></p>

<p><sub>2. a Youtube series he prodouced on rational expectations <br>
<a href="https://www.youtube.com/watch?v=hMaJ0CMRNlk&amp;list=PL1BUBmE-wKq_McDWvrAsiMygpupxOpvpg" rel="nofollow noreferrer">https://www.youtube.com/watch?v=hMaJ0CMRNlk&amp;list=PL1BUBmE-wKq_McDWvrAsiMygpupxOpvpg</a> </sub></p>

CC BY-SA 3.0
2018-02-23T19:24:40.777Z
This is a good question as it touches on some important topics. IMO, I don't think this answer is quite right and I have a few objections: (1) I don't know what you mean by "validity of ARIMA." I don't think this concept is well defined in this context. It might help if you clarify what you mean by "validity."
jmbejara
CC BY-SA 3.0
2018-02-23T19:27:38.590Z
(2) I generally don't agree with the statement that its "validity is based on the assumption that markets are efficient." ARIMA is simply a flexible statistical model. Under some assumptions, rational expectations puts restrictions on the parameters of the model. These restrictions can tested formally (hypothesis testing). Hypothesis testing is a slightly different endeavour than forecasting. For simple forecasting, a flexible model like ARIMA may be sufficient and adding in the restrictions that rational expectations might impose won't clearly improve your forecasting.
jmbejara
CC BY-SA 3.0
2018-02-23T19:31:48.830Z
(3) You write that $y_t=\tilde{y}_t+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$ is your efficient markets model. This isn't the case. Even in the book you references, this is references as "a model that satisfies the efficient markets condition." This itself is even a little out of date, but that's besides the point. I would be careful to not claim that rational expectations is more than it is---that is, that economic actors are endowed with beliefs about the underlying statistical model that match the true model.
jmbejara
CC BY-SA 3.0
2018-02-23T19:32:59.950Z
(4) Also, the model that is used in the book is not necessarily univariate. Market expectations in $X^e_{\tau}$ may be and probably are formed on other variables besides $X$. See p. 15 of the cited book.
jmbejara
CC BY-SA 3.0
2018-02-23T19:34:16.163Z
I think that @AndrewM 's answer covers things pretty well.
jmbejara
CC BY-SA 3.0
2018-02-23T19:57:11.570Z
@jmbejara (1) what I mean by the validity of  ARIMA models is that we assume the variable in question follows its own independent DGP unaffected by other variables. This seems rather out of touch with the whole approach structural models have taken by being based in theory of variables interacting with each other. On an outset the use of atheoretical ARIMA modelling is out of line with economic theory.
EconJohn
CC BY-SA 3.0
2018-02-23T20:04:30.640Z
@jmbejara (2) All I'm saying is that if RE holds all that is necessary in economic modelling is its own lags. If all information is used by economic agents univariate autoregressive equations are all you need (save some seasonal effects).
EconJohn
CC BY-SA 3.0
2018-02-23T20:07:31.830Z
@jmbejara (3) So you  take issue with  the name? the book title calls them efficient market models. (4) thats only if neutrality constraints are relaxed. I'm arguing from the fact they are still held.
EconJohn
CC BY-SA 3.0
2018-02-23T20:08:02.277Z
Re (1): The appropriate interpretation seems to be "is ARIMA useful for forecasting?" The answer, at least in-sample and absent major policy shifts, seems to be yes. Note also that Mishkin says (on p. 21) that "it is difficult on theoretical grounds to exclude any piece of information available at time t - 1" in forming $X^e_{t}$ and that "This suggests that an atheoretical statistical procedure may be superior to economic theory for deciding on the forecasting equation's specification."
jmbejara
CC BY-SA 3.0
2018-02-23T20:09:23.800Z
Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/73607/discussion-between-jmbejara-and-econjohn).
jmbejara
CC BY-SA 3.0
2018-02-11T01:57:53.993Z
2018-03-07T22:07:55.537Z
2018-03-07T22:07:55.537Z
EconJohn
<p>So your question is basically asking why a purely statistical model would be better at forecasting than a model based on theoretical relationships. For example, think about a simple form of the ARIMA model i.e. an AR model where you're just using lags of the process to explain the process itself. Now think about what is contained in the lags of the process... If the true process (i.e. the DGP) is determined by theoretical relationships between other variables then these will automatically be included in the lag. So the ARIMA model is a statistical way of modeling the process to make sure you capture the statistical properties of the underlying relationship but does not necessarily provide an explanation for what is going on. In forecasting, especially when series are volatile and erratic, you often find that simple statistical models such as these do a better job since the underlying relationships may not be very stable or may be more complicated than any theory we can come up with. This is related to a recent article  which you may find interesting: <a href="http://www.sciencedirect.com/science/article/pii/S0169207017300997" rel="nofollow noreferrer">http://www.sciencedirect.com/science/article/pii/S0169207017300997</a></p>

CC BY-SA 3.0
2017-11-02T15:58:14.700Z
Wow there is a lot of smack talk in this paper.
EconJohn
CC BY-SA 3.0
2017-11-02T08:54:08.670Z
2017-11-02T08:54:08.670Z
Andrew M
<p>In short, for the purposes of forecasting, it is often convenient and even beneficial to ignore economic theory. The restrictions implied by economic theory is necessary for inference when the question of interest is causal in nature (e.g., when you want to understand the underlying causal relationships). However, if the forecast you're interested is more of the variety of a sophisticated extrapolation, then something simple like ARIMA may be more helpful and more accurate.</p>

<p>I think the following are helpful references. </p>

<h2>Why would I not be interested in the underlying causal relationship?</h2>

<p>Consider the following example from "Introduction to Econometrics" (3rd Edition), by Stock and Watson (p. 517).</p>

<blockquote>
  <p>The simplest regression model in Chapter 3 related test scores to the
  student-teacher ration (STR): \begin{equation} \widehat {Test Score} = 989.9 - 2.28 \times STR \tag{14.1} \end{equation} As was discussed in Chapter 6, a school superintendent, contemplating hiring more teachers
  to reduce class sizes, would not consider this equation to be very
  helpful. The estimated slope coefficient in Equation (14.1) fails to
  provide a useful estimate of the causal effect on test scores of the
  student-teacher ratio because of probable omitted variable bias
  arising from omission of school and student characteristics that are
  determinant of test scores and that are correlated with the
  student-teacher ratio.</p>
  
  <p>In contrast, as was discussed in Chapter 9, a parent who is
  considering moving to a school district might find Equation (14.1)
  more helpful. Even though the coefficient does not have a causal
  interpretation, the regression could help the parent forecast test
  scores in a district for which they are not publicly available. More
  generally, a regression model can be useful for forecasting even if
  none of its coefficients has a causal interpretation. From the
  perspective of forecasting, what is important is that the model
  provides as accurate a forecast as possible.</p>
</blockquote>

<p>Along these same lines, this is a point that is brought up frequently
when discussing Machine Learning. Machine Learning may not be so useful in helping to understand underlying causal relationships. However, often we are more interested in the predictions/forecasts. <a href="https://www.youtube.com/watch?v=xl3yQBhI6vY" rel="nofollow noreferrer">Sendhil Mullainathan in this talk</a> on "Machine Learning and Prediction in Economics and Finance" gives the useful "rule of thumb" rubric---are you more interested in the $\hat y$ or the $\hat beta$?</p>

<h2>Why would something like ARIMA forecast better than a model that better reflected economic theory?</h2>

<p>Consider this useful passage from "A Guide to Econometrics" (6th Edition) by Peter Kennedy (p. 333). </p>

<blockquote>
  <p>The main competitors to econometric models for forecasting purposes
  are Box-Jenkins, or ARIMA (autoregressive integrated moving average),
  models explained in some detail in chapter 19. Univariate Box-Jenkins
  models are sophisticated extrapolation methods, using only past values
  of the variable being forecast to generate forecasts; they ignore the
  many explanatory variables that form the foundation of econometric
  models. There are several reasons why forecasters should be interested
  in these naive models: thanks to improved computer software, they are
  easy and cheap to produce; the extra information required to estimate
  a proper econometric model may be expen­sive to obtain ; forecasts
  from such models can serve as a useful benchmark for com­parison
  purposes; forecasts from this process can be combined with other
  forecasts to produce improved forecasts ; and they are useful as a
  preliminary step for further modeling---they clarify the nature of the
  data and make clear what behavior patterns require explanation.</p>
  
  <p>During the 1970s controversy raged over the relative forecasting
  merits of econo­metric models and ARIMA models, prompted by studies
  claiming the superiority of the ARIMA models. As noted in chapter 19,
  this led to a synthesis of the two approaches, and prompted the
  development of models, such as error-correction models (ECMs), which
  paid more attention to dynamics. In retro spect, the reason why
  econometric models performed so poorly in these comparisons was
  because of misspecification errors in the econometric models,
  primarily with respect to their dynamic structure. It is generally
  acknowledged that whenever specification or conditioning errors
  ren­der econometric models impractical (which some claim is most of
  the time), the Box-Jenkins approach has considerable merit for
  forecasting. It is also recognized that if an econometric model is
  outperformed by an ARIMA model, this is evidence that the econometric
  model is misspecified.</p>
</blockquote>

<h2>Can a univariate model like ARIMA represent a rational expectations equilibrium?</h2>

<p>Not really. Most economics theory employ rational expectations that lead to a nonlinear system of equations (in multiple variables) that describe the model's dynamics. Often, these are approximated (e.g., log-linear approximation), so that they can be solved and estimated using something like a <a href="https://lectures.quantecon.org/py/linear_models.html" rel="nofollow noreferrer">linear state space model</a>---but many still require multiple variables (here, arranged in a vector) to describe the full dimensionality of the model. For an example of one particular model that uses rational expectations, see p. 30 of "Recursive Models of Dynamic Linear Economics" by Hansen and Sargent. </p>

<blockquote>
  <p>Consider a stochastic process $\{p_t\}$ related to a stochastic process $\{m_t\}$ via 
  \begin{equation}
p_t = \lambda E_t p_{t+1} + \gamma m_t \tag{(2.4.38)}
\end{equation}
  where
  \begin{equation}
m_t = G x_t \tag{(2.4.39)}
\end{equation}
  and $x_t$ is governed by
  $$
x_{t+1} = A x_t + C w_{t+1}, \text{ for } t = 0,1,2,...
$$</p>
  
  <p>... 
  Collecting results, we have that $(p_t, m_t)$ satisfies
  \begin{align*}
\begin{bmatrix}{p_t \\ m_t }\end{bmatrix} 
&amp;= 
\begin{bmatrix}
\gamma G(I - \lambda A)^{-1} \\ G
\end{bmatrix}
x_t
\\
x_{t+1} &amp;= A x_t + C w_{t+1}.
\end{align*}
  [This system] embodies the cross-equation restrictions associated with
  rational expectations models: note that the same parameters in $A,G$ that pin > down the stochastic process for $m_t$ also enter the equation that determines > $p_t$ as a function of the state $x_t$.</p>
</blockquote>

<p>This equation, written flexibly, allows for many state variables and many (possibly orthogonal) shocks. As such, this cannot be represented by a simple univariate model like ARIMA.</p>

<p>Many other examples can be found in "Structural Econometrics" by Dejong and Dave or more in "Recursive Models of Dynamic Linear Economics", by Hansen and Sargent. Some will only have a single dimensional state space. Some will have much larger state spaces.</p>

<p>However, again, describing the model in full may not be necessary if you're only interested in simple forecasting. Hence, something like ARIMA is an attractive option.</p>

CC BY-SA 3.0
2018-02-26T21:57:38.940Z
I'll read this tonight. looks very comprehensive
EconJohn
CC BY-SA 3.0
2018-02-26T23:46:32.780Z
It would seem that the source to support the argument that AR(p) models are not a result of  RE equilibrium is a refutation for your own position. See the structural law of motion at the end of https://lectures.quantecon.org/py/rational_expectations.html
EconJohn
CC BY-SA 3.0
2018-02-27T00:44:59.957Z
@EconJohn This doesn't refute it. The law of motion, the one for aggregate productivity, that you point out simply has a single shock and single state variable. This could easily be extend to include more states and more shocks. I have edited my answer to include an example that more clearly illustrates this. Most models you see in modern macro textbooks are RE models. However, most of them don't explicitly state that they use RE. The examples I've chosen are those that both say that they are RE and exhibit (or could exhibit) multiple irreducible dimensions.  Hope this helps!
jmbejara
CC BY-SA 3.0
2018-02-26T21:18:10.860Z
2018-02-27T00:40:17.220Z
2018-02-27T00:40:17.220Z
jmbejara
<p>Ive noticed the use of univariate forecasting methods (i.e the ARIMA model) employed in forecasting inflation (see references below).</p>

<p>How is this a valid approach in terms of forecasting economic variables? does this approach not omit the large amount of economic theory which determines how these variables are determined (i.e via equilibrium and interaction of variables other than themselves?)</p>

<hr>

<p><sub>1.<a href="https://scholar.harvard.edu/files/stock/files/forecastinginflation.pdf" rel="nofollow noreferrer">https://scholar.harvard.edu/files/stock/files/forecastinginflation.pdf</a> see pages 6-7<br>2.<a href="http://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdfhttp://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdf" rel="nofollow noreferrer">http://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdfhttp://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdf</a> <br>
3.<a href="http://www.unagaliciamoderna.com/eawp/coldata/upload/pakistans%20inflaction_arima%20model.pdf" rel="nofollow noreferrer">http://www.unagaliciamoderna.com/eawp/coldata/upload/pakistans%20inflaction_arima%20model.pdf</a> </sub></p>

<macroeconomics><econometrics><reference-request>
How is the ARIMA model a valid approach for forecasting Economic Variables?
EconJohn
# How is the ARIMA model a valid approach for forecasting Economic Variables?

(Asked by: username_0 on 2017-11-01)

Ive noticed the use of univariate forecasting methods (i.e the ARIMA model) employed in forecasting inflation (see references below).

How is this a valid approach in terms of forecasting economic variables? does this approach not omit the large amount of economic theory which determines how these variables are determined (i.e via equilibrium and interaction of variables other than themselves?)

---

1.<https://scholar.harvard.edu/files/stock/files/forecastinginflation.pdf> see pages 6-7  
2.<http://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdfhttp://repository.graduateinstitute.ch/record/294965/files/HEIDWP05-2017.pdf>   
3.<http://www.unagaliciamoderna.com/eawp/coldata/upload/pakistans%20inflaction_arima%20model.pdf>

**Tags:** macroeconomics, econometrics, reference-request

---

## 3 answers

### Answer by username_0 on 2018-02-11. Score: -1 | Accepted answer

I came across the answer to the validity of univariate modelling in economics (albet indirectly) from the book called [A Rational Expectations Approach to Macroeconometrics](http://press.uchicago.edu/ucp/books/book/chicago/R/bo5947912.html). This answer is based on the material in the book, for a better understanding I reccomend actually going through it.

> In short the validity of ARIMA models is based on the assumption that the market is always in equilibrium or **efficient**1.

In order to understand how use of ARIMA models in Econometrics is a valid way to forecast economic variables, we must understand models which considers the case of for **rational expectations** (other wise referred to as **efficient markets**).

your standard efficient markets model is:

$$y\_t=\tilde{y}\_t+\sum\_{i=0}^N\beta\_i(X\_{t-i}-X\_{t-i}^e)+\epsilon\_t$$
where:

* $y\_t=$ Economic variable of interest at time $t$ like unemployment or real output.
* $\tilde{y}\_t=$Natural rate or equilibrium of economic variable at time $t$
* $\beta\_i=$ coefficents.
* $X\_{t-i}=$ an aggreagte demand variable like money growth, Inflation or Nominal GDP growth.
* $X^e\_{t-i}=$ Anticipated aggregate demand conditional on knowing all information at point $t-i$
* $\epsilon\_t=$ an error term

This model accounts for variability that can be present when a market does not have all information pertaining to $y\_t$.

combining this formula using a specification used with the Lucas supply function:
$$\tilde{y}=\sum\_{i=1}^N\lambda\_iy\_{t-i}$$

combining the two equations we get:

$$y\_t=\sum\_{i=1}^N\lambda\_iy\_{t-i}+\sum\_{i=0}^N\beta\_i(X\_{t-i}-X\_{t-i}^e)+\epsilon\_t$$

This model tells us that if rational expectations hold,the relationship of $y\_t=\sum\_{i=1}^N\lambda\_iy\_{t-i}$, thus providing validity for the use of uni variate models in economics.

For more reading on this topic, below is posted links to the three chapters which helped me:  
1)<http://www.nber.org/chapters/c10241.pdf>  
2)<http://www.nber.org/chapters/c10243.pdf>  
3)<http://www.nber.org/chapters/c10244.pdf>

**UPDATE:** Upon Contacting [Dr. Paul Middleditch on Twitter](https://twitter.com/MyProgDirector/status/971261126657888256) he says that the RHS variables must be replaced with a proxy which is normally the value of the previous period. ARIMA methodology satisfies such constraints.2

---

1. Mathematically this statement can be written as:
$$\mathbb{E}\_m(X\_t|\phi\_{t-1})=\mathbb{E}(X\_t|\phi\_{t-1})$$

where the expected value given by the market of $X\_t$ conditional on $\phi\_{t-1}$ the set of all available information at time $t-1$, is equal to the true expected value of $X\_t$

It follows logically that $\mathbb{E}[X\_t-\mathbb{E}\_m(X\_t)|\phi\_{t-1}]=0$

2. a Youtube series he prodouced on rational expectations   
<https://www.youtube.com/watch?v=hMaJ0CMRNlk&list=PL1BUBmE-wKq_McDWvrAsiMygpupxOpvpg>

> **Comments:**
> **username_1**: This is a good question as it touches on some important topics. IMO, I don't think this answer is quite right and I have a few objections: (1) I don't know what you mean by "validity of ARIMA." I don't think this concept is well defined in this context. It might help if you clarify what you mean by "validity."  
> **username_1**: (2) I generally don't agree with the statement that its "validity is based on the assumption that markets are efficient." ARIMA is simply a flexible statistical model. Under some assumptions, rational expectations puts restrictions on the parameters of the model. These restrictions can tested formally (hypothesis testing). Hypothesis testing is a slightly different endeavour than forecasting. For simple forecasting, a flexible model like ARIMA may be sufficient and adding in the restrictions that rational expectations might impose won't clearly improve your forecasting.  
> **username_1**: (3) You write that $y_t=\tilde{y}_t+\sum_{i=0}^N\beta_i(X_{t-i}-X_{t-i}^e)+\epsilon_t$ is your efficient markets model. This isn't the case. Even in the book you references, this is references as "a model that satisfies the efficient markets condition." This itself is even a little out of date, but that's besides the point. I would be careful to not claim that rational expectations is more than it is---that is, that economic actors are endowed with beliefs about the underlying statistical model that match the true model.  
> **username_1**: (4) Also, the model that is used in the book is not necessarily univariate. Market expectations in $X^e_{\tau}$ may be and probably are formed on other variables besides $X$. See p. 15 of the cited book.  
> **username_1**: I think that @AndrewM 's answer covers things pretty well.  
> **username_0**: @jmbejara (1) what I mean by the validity of  ARIMA models is that we assume the variable in question follows its own independent DGP unaffected by other variables. This seems rather out of touch with the whole approach structural models have taken by being based in theory of variables interacting with each other. On an outset the use of atheoretical ARIMA modelling is out of line with economic theory.  
> **username_0**: @jmbejara (2) All I'm saying is that if RE holds all that is necessary in economic modelling is its own lags. If all information is used by economic agents univariate autoregressive equations are all you need (save some seasonal effects).  
> **username_0**: @jmbejara (3) So you  take issue with  the name? the book title calls them efficient market models. (4) thats only if neutrality constraints are relaxed. I'm arguing from the fact they are still held.  
> **username_1**: Re (1): The appropriate interpretation seems to be "is ARIMA useful for forecasting?" The answer, at least in-sample and absent major policy shifts, seems to be yes. Note also that Mishkin says (on p. 21) that "it is difficult on theoretical grounds to exclude any piece of information available at time t - 1" in forming $X^e_{t}$ and that "This suggests that an atheoretical statistical procedure may be superior to economic theory for deciding on the forecasting equation's specification."  
> **username_1**: Let us [continue this discussion in chat](http://chat.stackexchange.com/rooms/73607/discussion-between-jmbejara-and-econjohn).  

---

### Answer by username_2 on 2017-11-02. Score: 3

So your question is basically asking why a purely statistical model would be better at forecasting than a model based on theoretical relationships. For example, think about a simple form of the ARIMA model i.e. an AR model where you're just using lags of the process to explain the process itself. Now think about what is contained in the lags of the process... If the true process (i.e. the DGP) is determined by theoretical relationships between other variables then these will automatically be included in the lag. So the ARIMA model is a statistical way of modeling the process to make sure you capture the statistical properties of the underlying relationship but does not necessarily provide an explanation for what is going on. In forecasting, especially when series are volatile and erratic, you often find that simple statistical models such as these do a better job since the underlying relationships may not be very stable or may be more complicated than any theory we can come up with. This is related to a recent article which you may find interesting: <http://www.sciencedirect.com/science/article/pii/S0169207017300997>

> **Comments:**
> **username_0**: Wow there is a lot of smack talk in this paper.  

---

### Answer by username_1 on 2018-02-26. Score: 3

In short, for the purposes of forecasting, it is often convenient and even beneficial to ignore economic theory. The restrictions implied by economic theory is necessary for inference when the question of interest is causal in nature (e.g., when you want to understand the underlying causal relationships). However, if the forecast you're interested is more of the variety of a sophisticated extrapolation, then something simple like ARIMA may be more helpful and more accurate.

I think the following are helpful references.

## Why would I not be interested in the underlying causal relationship?

Consider the following example from "Introduction to Econometrics" (3rd Edition), by Stock and Watson (p. 517).

> The simplest regression model in Chapter 3 related test scores to the
> student-teacher ration (STR): \begin{equation} \widehat {Test Score} = 989.9 - 2.28 \times STR \tag{14.1} \end{equation} As was discussed in Chapter 6, a school superintendent, contemplating hiring more teachers
> to reduce class sizes, would not consider this equation to be very
> helpful. The estimated slope coefficient in Equation (14.1) fails to
> provide a useful estimate of the causal effect on test scores of the
> student-teacher ratio because of probable omitted variable bias
> arising from omission of school and student characteristics that are
> determinant of test scores and that are correlated with the
> student-teacher ratio.
>
> In contrast, as was discussed in Chapter 9, a parent who is
> considering moving to a school district might find Equation (14.1)
> more helpful. Even though the coefficient does not have a causal
> interpretation, the regression could help the parent forecast test
> scores in a district for which they are not publicly available. More
> generally, a regression model can be useful for forecasting even if
> none of its coefficients has a causal interpretation. From the
> perspective of forecasting, what is important is that the model
> provides as accurate a forecast as possible.

Along these same lines, this is a point that is brought up frequently
when discussing Machine Learning. Machine Learning may not be so useful in helping to understand underlying causal relationships. However, often we are more interested in the predictions/forecasts. [Sendhil Mullainathan in this talk](https://www.youtube.com/watch?v=xl3yQBhI6vY) on "Machine Learning and Prediction in Economics and Finance" gives the useful "rule of thumb" rubric---are you more interested in the $\hat y$ or the $\hat beta$?

## Why would something like ARIMA forecast better than a model that better reflected economic theory?

Consider this useful passage from "A Guide to Econometrics" (6th Edition) by Peter Kennedy (p. 333).

> The main competitors to econometric models for forecasting purposes
> are Box-Jenkins, or ARIMA (autoregressive integrated moving average),
> models explained in some detail in chapter 19. Univariate Box-Jenkins
> models are sophisticated extrapolation methods, using only past values
> of the variable being forecast to generate forecasts; they ignore the
> many explanatory variables that form the foundation of econometric
> models. There are several reasons why forecasters should be interested
> in these naive models: thanks to improved computer software, they are
> easy and cheap to produce; the extra information required to estimate
> a proper econometric model may be expen­sive to obtain ; forecasts
> from such models can serve as a useful benchmark for com­parison
> purposes; forecasts from this process can be combined with other
> forecasts to produce improved forecasts ; and they are useful as a
> preliminary step for further modeling---they clarify the nature of the
> data and make clear what behavior patterns require explanation.
>
> During the 1970s controversy raged over the relative forecasting
> merits of econo­metric models and ARIMA models, prompted by studies
> claiming the superiority of the ARIMA models. As noted in chapter 19,
> this led to a synthesis of the two approaches, and prompted the
> development of models, such as error-correction models (ECMs), which
> paid more attention to dynamics. In retro spect, the reason why
> econometric models performed so poorly in these comparisons was
> because of misspecification errors in the econometric models,
> primarily with respect to their dynamic structure. It is generally
> acknowledged that whenever specification or conditioning errors
> ren­der econometric models impractical (which some claim is most of
> the time), the Box-Jenkins approach has considerable merit for
> forecasting. It is also recognized that if an econometric model is
> outperformed by an ARIMA model, this is evidence that the econometric
> model is misspecified.

## Can a univariate model like ARIMA represent a rational expectations equilibrium?

Not really. Most economics theory employ rational expectations that lead to a nonlinear system of equations (in multiple variables) that describe the model's dynamics. Often, these are approximated (e.g., log-linear approximation), so that they can be solved and estimated using something like a [linear state space model](https://lectures.quantecon.org/py/linear_models.html)---but many still require multiple variables (here, arranged in a vector) to describe the full dimensionality of the model. For an example of one particular model that uses rational expectations, see p. 30 of "Recursive Models of Dynamic Linear Economics" by Hansen and Sargent.

> Consider a stochastic process $\{p\_t\}$ related to a stochastic process $\{m\_t\}$ via
> \begin{equation}
> p\_t = \lambda E\_t p\_{t+1} + \gamma m\_t \tag{(2.4.38)}
> \end{equation}
> where
> \begin{equation}
> m\_t = G x\_t \tag{(2.4.39)}
> \end{equation}
> and $x\_t$ is governed by
> $$
> x\_{t+1} = A x\_t + C w\_{t+1}, \text{ for } t = 0,1,2,...
> $$
>
> ...
> Collecting results, we have that $(p\_t, m\_t)$ satisfies
> \begin{align\*}
> \begin{bmatrix}{p\_t \\ m\_t }\end{bmatrix}
> &=
> \begin{bmatrix}
> \gamma G(I - \lambda A)^{-1} \\ G
> \end{bmatrix}
> x\_t
> \\
> x\_{t+1} &= A x\_t + C w\_{t+1}.
> \end{align\*}
> [This system] embodies the cross-equation restrictions associated with
> rational expectations models: note that the same parameters in $A,G$ that pin > down the stochastic process for $m\_t$ also enter the equation that determines > $p\_t$ as a function of the state $x\_t$.

This equation, written flexibly, allows for many state variables and many (possibly orthogonal) shocks. As such, this cannot be represented by a simple univariate model like ARIMA.

Many other examples can be found in "Structural Econometrics" by Dejong and Dave or more in "Recursive Models of Dynamic Linear Economics", by Hansen and Sargent. Some will only have a single dimensional state space. Some will have much larger state spaces.

However, again, describing the model in full may not be necessary if you're only interested in simple forecasting. Hence, something like ARIMA is an attractive option.

> **Comments:**
> **username_0**: I'll read this tonight. looks very comprehensive  
> **username_0**: It would seem that the source to support the argument that AR(p) models are not a result of  RE equilibrium is a refutation for your own position. See the structural law of motion at the end of https://lectures.quantecon.org/py/rational_expectations.html  
> **username_1**: @EconJohn This doesn't refute it. The law of motion, the one for aggregate productivity, that you point out simply has a single shock and single state variable. This could easily be extend to include more states and more shocks. I have edited my answer to include an example that more clearly illustrates this. Most models you see in modern macro textbooks are RE models. However, most of them don't explicitly state that they use RE. The examples I've chosen are those that both say that they are RE and exhibit (or could exhibit) multiple irreducible dimensions.  Hope this helps!  

---

