<p>You got it quite nicely:</p>

<ul>
<li>An AR(1) process would imply population in $t$ depends linearly on the population level at $t-1$</li>
<li>An MA(1) process would imply population in $t$ depends linearly on the population shock, event or "surprise" at $t-1$.</li>
<li>An I(1) process would imply population growth (that is, the first difference, $X_t-X_{t-1}$) is a stationary process and doesn't have a trend; the population level itself isn't. In the real world, though, it's strange to find an I(2) model, much less I(>2).</li>
</ul>

<p>Mix everything up and you can forecast population with a combination of previous population levels, surprising events and population growths.</p>

CC BY-SA 3.0
2017-11-28T21:49:01.240Z
How come I(1) is supposed to imply population growth has a trend? E.g. ARIMA(0,1,0) is a random walk, hence no trend in growth, but a stochastic trend in level. @EconJohn
Richard Hardy
CC BY-SA 3.0
2017-11-30T18:43:22.440Z
Use a random walk and calculate the first difference (growth). Check that it has a trend (likely, the mean is 0).
one_teach_wonder
CC BY-SA 3.0
2017-12-02T16:15:12.860Z
This is incorrect. Check the definition of a random walk in, say, [Wikipedia](https://en.wikipedia.org/wiki/Random_walk).
Richard Hardy
CC BY-SA 3.0
2017-12-03T18:57:24.073Z
Strange that you didn't cite Resnick or Hamilton, but since you seem to like Wikipedia, look here: https://en.wikipedia.org/wiki/Order_of_integration
one_teach_wonder
CC BY-SA 3.0
2017-12-03T19:26:40.413Z
I don't see how this supports your statement or invalidates mine. Could you please be more specific? Regarding references, I stick to what is the easiest to find. I could go to my shelf and open Hamilton (I do not use Resnick), but that would take longer than necessary.
Richard Hardy
CC BY-SA 3.0
2017-12-03T19:36:26.770Z
"In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process." So, I(1) means taking a single difference yields a stationary process.
one_teach_wonder
CC BY-SA 3.0
2017-12-03T19:44:45.453Z
I agree with you (and Wikipedia) completely on this. However, read my comment above carefully. *How come I(1) is supposed to imply population growth has a trend? E.g. ARIMA(0,1,0) is a random walk, hence no trend in growth, but a stochastic trend in level.* Also, your claim in the next comment is incorrect, because neither a random walk nor its first differences do not have a deterministic trend by the definition of the random walk.
Richard Hardy
CC BY-SA 3.0
2017-12-03T20:03:46.803Z
Oh, now I see it, I interchanged the terms "stationary process" and "trend". That's what I get for reading the course materials in two languages, I guess. Indeed, a random walk has a trend/isn't a stationary process and the first difference hasn't/is. My bad, I'll correct it right away.
one_teach_wonder
CC BY-SA 3.0
2017-12-03T21:16:19.510Z
Great! I am glad we found the root of the confusion.
Richard Hardy
CC BY-SA 3.0
2018-02-23T19:57:33.550Z
The fact that the AR model has an equivalent MA representation makes interpreting these coefficients this way a little suspect. See the answer by @AndrewM
jmbejara
CC BY-SA 3.0
2017-11-27T19:46:58.007Z
2017-12-03T20:15:05.157Z
2017-12-03T20:15:05.157Z
one_teach_wonder
<p>So we can think of an AR(1) process as:</p>

<p>$y_t = a*y_{t-1} +e_t$</p>

<p>where $e_t$ is some shock / noise process that we can't explain. Note that we can rewrite an AR process as an MA process where by tracing through we get</p>

<p>$y_t = a^2*y_{t-2}+a*e_{t-1}+e_t\approx \sum_{i=0}^{t-1}a^i*e_{t-i}$</p>

<p>So that an AR process is just the weighted sum of all previous shocks where $a$ is what determines the persistence of past shocks. </p>

<p>So then what does it mean to say we have an AR(1) process? Well it means that the outcome today is to some extent dependent on the outcome yesterday (as you yourself suggested). In an extreme case we can set $a=1$ so that we have a random walk process where</p>

<p>$\Delta y_t=y_t-y_{t-1}=e_t$</p>

<p>This implies that the outcome today depends entirely on the outcome yesterday and any additional change in the series is purely due to random noise. Alternatively if $a\geq1$ we would say the process is non-stationary or explosive while if $a&lt;1$ then the process is stationary around the mean of the process or of the shocks themselves.</p>

<p>An MA process has a similar interpretation except we think of it more in terms of unexplainable shocks or noise with some degree of persistence. The important thing is that it is not really something we have a variable for in the MA case but rather this notion that the the process just depends on this noise component. It is more common to find these types of models for forecasting in finance where there's alot of movement but a less clear sense of what might be driving it.</p>

<p>In terms of population forecasting, the AR component could definitely be interpreted as you suggested, while the MA component is less easy to interpret. It's really just a sum of shocks that are otherwise unexplained. If you had an explanation for it (i.e. immigration etc.) then you would likely want to include that as an explanatory variable to model it explicitly rather than approximately through the MA process.</p>

CC BY-SA 3.0
2018-02-23T19:55:46.990Z
This is a good answer. I just wanted to add the following. You could give the AR component the interpretation you suggest, but that doesn't mean that the AR coefficient you estimate will recover the parameter with this meaning. The act of giving the AR component this interpretation amounts to constructing a structural equation and causal model. The question of whether your estimation procedure recovers parameters with this interpretation is another story. The fact that the AR model has an equivalent MA interpretation makes this point of view relatively easy to see.
jmbejara
CC BY-SA 3.0
2017-11-27T18:27:25.393Z
2017-11-27T18:27:25.393Z
Andrew M
<p>ARIMA models, from what I know, are atheoretical models in the sense that they don't provide us with meaningful economic interpretation of a process.</p>
<p>However in population forecasting, can ARIMA models be interpreted in a meaningful way?</p>
<p>i.e.
<br></p>
<ul>
<li>can <span class="math-container">$AR(1)$</span> process can be interpreted as:
<blockquote>
<p><em>population growth in period <span class="math-container">$t$</span> is a function of population in period <span class="math-container">$t-1$</span></em></p>
</blockquote>
</li>
<li>can <span class="math-container">$MA(1)$</span> process can be interpreted as:
<blockquote>
<p><em>population in period <span class="math-container">$t$</span> is a function of a policy or ongoing event (like immigration or emigration) which has occured in <span class="math-container">$t-1$</span>.</em></p>
</blockquote>
</li>
</ul>

<econometrics><demographics>
ARIMA modelling and Economic interpretation
EconJohn
# ARIMA modelling and Economic interpretation

(Asked by: username_0 on 2017-11-26)

ARIMA models, from what I know, are atheoretical models in the sense that they don't provide us with meaningful economic interpretation of a process.

However in population forecasting, can ARIMA models be interpreted in a meaningful way?

i.e.

* can $AR(1)$ process can be interpreted as:
  > *population growth in period $t$ is a function of population in period $t-1$*
* can $MA(1)$ process can be interpreted as:
  > *population in period $t$ is a function of a policy or ongoing event (like immigration or emigration) which has occured in $t-1$.*

**Tags:** econometrics, demographics

---

## 2 answers

### Answer by username_1 on 2017-11-27. Score: 1 | Accepted answer

You got it quite nicely:

* An AR(1) process would imply population in $t$ depends linearly on the population level at $t-1$
* An MA(1) process would imply population in $t$ depends linearly on the population shock, event or "surprise" at $t-1$.
* An I(1) process would imply population growth (that is, the first difference, $X\_t-X\_{t-1}$) is a stationary process and doesn't have a trend; the population level itself isn't. In the real world, though, it's strange to find an I(2) model, much less I(>2).

Mix everything up and you can forecast population with a combination of previous population levels, surprising events and population growths.

> **Comments:**
> **username_2**: How come I(1) is supposed to imply population growth has a trend? E.g. ARIMA(0,1,0) is a random walk, hence no trend in growth, but a stochastic trend in level. @EconJohn  
> **username_1**: Use a random walk and calculate the first difference (growth). Check that it has a trend (likely, the mean is 0).  
> **username_2**: This is incorrect. Check the definition of a random walk in, say, [Wikipedia](https://en.wikipedia.org/wiki/Random_walk).  
> **username_1**: Strange that you didn't cite Resnick or Hamilton, but since you seem to like Wikipedia, look here: https://en.wikipedia.org/wiki/Order_of_integration  
> **username_2**: I don't see how this supports your statement or invalidates mine. Could you please be more specific? Regarding references, I stick to what is the easiest to find. I could go to my shelf and open Hamilton (I do not use Resnick), but that would take longer than necessary.  
> **username_1**: "In other words, a process is integrated to order d if taking repeated differences d times yields a stationary process." So, I(1) means taking a single difference yields a stationary process.  
> **username_2**: I agree with you (and Wikipedia) completely on this. However, read my comment above carefully. *How come I(1) is supposed to imply population growth has a trend? E.g. ARIMA(0,1,0) is a random walk, hence no trend in growth, but a stochastic trend in level.* Also, your claim in the next comment is incorrect, because neither a random walk nor its first differences do not have a deterministic trend by the definition of the random walk.  
> **username_1**: Oh, now I see it, I interchanged the terms "stationary process" and "trend". That's what I get for reading the course materials in two languages, I guess. Indeed, a random walk has a trend/isn't a stationary process and the first difference hasn't/is. My bad, I'll correct it right away.  
> **username_2**: Great! I am glad we found the root of the confusion.  
> **username_3**: The fact that the AR model has an equivalent MA representation makes interpreting these coefficients this way a little suspect. See the answer by @AndrewM  

---

### Answer by username_4 on 2017-11-27. Score: 2

So we can think of an AR(1) process as:

$y\_t = a\*y\_{t-1} +e\_t$

where $e\_t$ is some shock / noise process that we can't explain. Note that we can rewrite an AR process as an MA process where by tracing through we get

$y\_t = a^2\*y\_{t-2}+a\*e\_{t-1}+e\_t\approx \sum\_{i=0}^{t-1}a^i\*e\_{t-i}$

So that an AR process is just the weighted sum of all previous shocks where $a$ is what determines the persistence of past shocks.

So then what does it mean to say we have an AR(1) process? Well it means that the outcome today is to some extent dependent on the outcome yesterday (as you yourself suggested). In an extreme case we can set $a=1$ so that we have a random walk process where

$\Delta y\_t=y\_t-y\_{t-1}=e\_t$

This implies that the outcome today depends entirely on the outcome yesterday and any additional change in the series is purely due to random noise. Alternatively if $a\geq1$ we would say the process is non-stationary or explosive while if $a<1$ then the process is stationary around the mean of the process or of the shocks themselves.

An MA process has a similar interpretation except we think of it more in terms of unexplainable shocks or noise with some degree of persistence. The important thing is that it is not really something we have a variable for in the MA case but rather this notion that the the process just depends on this noise component. It is more common to find these types of models for forecasting in finance where there's alot of movement but a less clear sense of what might be driving it.

In terms of population forecasting, the AR component could definitely be interpreted as you suggested, while the MA component is less easy to interpret. It's really just a sum of shocks that are otherwise unexplained. If you had an explanation for it (i.e. immigration etc.) then you would likely want to include that as an explanatory variable to model it explicitly rather than approximately through the MA process.

> **Comments:**
> **username_3**: This is a good answer. I just wanted to add the following. You could give the AR component the interpretation you suggest, but that doesn't mean that the AR coefficient you estimate will recover the parameter with this meaning. The act of giving the AR component this interpretation amounts to constructing a structural equation and causal model. The question of whether your estimation procedure recovers parameters with this interpretation is another story. The fact that the AR model has an equivalent MA interpretation makes this point of view relatively easy to see.  

---

