<p>Cochrane (1991) says that unit root tests have low power, so you need lots of data (at best) to distinguish among non-unit root alternatives:</p>

<blockquote>
  <p>This paper exploits the fact that any time series with a unit root can
  decomposed into a stationary series and a random walk. Since the
  random walk component can have arbitrarily small variance, tests for
  unit roots or trend stationarity have arbitrarily low power in finite
  samples. Furthermore, there are unit root processes whose likelihood
  functions and autocorrelation functions are arbitrarily close to
  those of any given stationary processes and vice versa, so there are
  stationary and unit root processes for which the result of any
  inference is arbitrarily close in finite samples.</p>
</blockquote>

<p><a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.308.7613&amp;rep=rep1&amp;type=pdf" rel="nofollow noreferrer">A critique of the application of unit root tests</a></p>

<p>However, depending on the test and the realistic alternative, this may be fine. David N. DeJong, John C. Nankervis, N.E.Savin, Charles H. Whiteman (1992): </p>

<blockquote>
  <p>Monte Carlo methods are used to study the size and
  power of serial-correlation-corrected versions of the Dickey-Fuller
  (1979,1981) unit root tests appropriate when the time series has
  unknown mean. The modifications do not cause serious size distortions
  or power deterioration in the white noise case. While studies in the
  literature have investigated the operating characteristics of these
  tests in the presence of moving average errors, of particular concern
  in this paper is the performance of these procedures in the presence
  of autoregressive errors. The Philips and Perron (1988) and Choi and
  Philips (1991) procedures are found to suffer from serious size
  distortions and have very low power when errors are autoregressively
  correlated. We conclude that even in the most favorable cases, these
  tests perform poorly against trend-stationary alternatives which are
  plausible for annual, quarterly, and monthly macroeconomic time
  series. The augmented Dickey-Fuller procedure, on the other hand, is
  reasonably well-behaved.</p>
</blockquote>

<p><a href="https://www.sciencedirect.com/science/article/pii/030440769290090E" rel="nofollow noreferrer">The power problems of unit root test in time series with autoregressive errors</a> </p>

<p>My impression is things have improved a bit in the power of more cutting edge estimators. See Haldrup and Jansson (2005)</p>

<blockquote>
  <p>A frequent criticism of unit root tests concerns the poor power and
  size properties that many of such tests exhibit. However, the past
  decade or so intensive research has been conducted to alleviate these
  problems and great advances have been made. The present paper provides
  a selective survey of recent contributions to improve upon both size
  and power of unit root tests and in so doing the approach of using
  rigorous statistical optimality criteria in the development of such
  tests is stressed. In addition to presenting tests where improved size
  can be achieved by modifying the standard Dickey-Fuller class of
  tests, the paper presents theory of optimal testing and the
  construction of power envelopes for unit root tests under different
  conditions allowing for serial correlation, deterministic components,
  assumptions regarding the initial condition, non-Gaussian errors, and
  the use of covariates.</p>
</blockquote>

<p><a href="https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1147594" rel="nofollow noreferrer">Improving Size and Power in the Unit Root Testing</a> </p>

<p>Nevertheless, I can't help thinking that 16 annual observations is not much data to discipline time series estimation, and there will likely be lots of somewhat plausible specifications you will not be able to reject. Instead, you could use economy theory to justify the functional form and the statistical tests you use. For example, in efficient markets (crudely), price levels are highly serially correlated (random walks) but returns are independent. So if you knew your data were in returns, you might just argue rather than test that they are not a unit root. </p>

CC BY-SA 4.0
2019-04-02T17:55:43.760Z
but is it okay if I will just use standard multiple ols regression for even if I have a small sample size?
valve01
CC BY-SA 4.0
2019-04-02T17:57:06.960Z
That's a different question that should be asked separately.
BKay
CC BY-SA 4.0
2019-04-02T17:47:01.130Z
2019-04-02T23:51:58.130Z
2019-04-02T23:51:58.130Z
BKay
<p>I have a 16 year time series (annual frequency with 16 observations). I will conduct an OLS regression. In this setting do I need a unit root test?</p>

<p>Do you have additional suggestions for things that I should do to pre-analyze my data? </p>

CC BY-SA 4.0
2019-04-02T10:36:11.020Z
make data stationary.
Jamzy
<macroeconomics><regression>
Are unit root tests necessary or useful on small samples of time series data?
valve01
# Are unit root tests necessary or useful on small samples of time series data?

(Asked by: username_0 on 2019-04-02)

I have a 16 year time series (annual frequency with 16 observations). I will conduct an OLS regression. In this setting do I need a unit root test?

Do you have additional suggestions for things that I should do to pre-analyze my data?

**Tags:** macroeconomics, regression
> **Comments:**
> **username_1**: make data stationary.  

---

## 1 answers

### Answer by username_2 on 2019-04-02. Score: 1 | Accepted answer

Cochrane (1991) says that unit root tests have low power, so you need lots of data (at best) to distinguish among non-unit root alternatives:

> This paper exploits the fact that any time series with a unit root can
> decomposed into a stationary series and a random walk. Since the
> random walk component can have arbitrarily small variance, tests for
> unit roots or trend stationarity have arbitrarily low power in finite
> samples. Furthermore, there are unit root processes whose likelihood
> functions and autocorrelation functions are arbitrarily close to
> those of any given stationary processes and vice versa, so there are
> stationary and unit root processes for which the result of any
> inference is arbitrarily close in finite samples.

[A critique of the application of unit root tests](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.308.7613&rep=rep1&type=pdf)

However, depending on the test and the realistic alternative, this may be fine. David N. DeJong, John C. Nankervis, N.E.Savin, Charles H. Whiteman (1992):

> Monte Carlo methods are used to study the size and
> power of serial-correlation-corrected versions of the Dickey-Fuller
> (1979,1981) unit root tests appropriate when the time series has
> unknown mean. The modifications do not cause serious size distortions
> or power deterioration in the white noise case. While studies in the
> literature have investigated the operating characteristics of these
> tests in the presence of moving average errors, of particular concern
> in this paper is the performance of these procedures in the presence
> of autoregressive errors. The Philips and Perron (1988) and Choi and
> Philips (1991) procedures are found to suffer from serious size
> distortions and have very low power when errors are autoregressively
> correlated. We conclude that even in the most favorable cases, these
> tests perform poorly against trend-stationary alternatives which are
> plausible for annual, quarterly, and monthly macroeconomic time
> series. The augmented Dickey-Fuller procedure, on the other hand, is
> reasonably well-behaved.

[The power problems of unit root test in time series with autoregressive errors](https://www.sciencedirect.com/science/article/pii/030440769290090E)

My impression is things have improved a bit in the power of more cutting edge estimators. See Haldrup and Jansson (2005)

> A frequent criticism of unit root tests concerns the poor power and
> size properties that many of such tests exhibit. However, the past
> decade or so intensive research has been conducted to alleviate these
> problems and great advances have been made. The present paper provides
> a selective survey of recent contributions to improve upon both size
> and power of unit root tests and in so doing the approach of using
> rigorous statistical optimality criteria in the development of such
> tests is stressed. In addition to presenting tests where improved size
> can be achieved by modifying the standard Dickey-Fuller class of
> tests, the paper presents theory of optimal testing and the
> construction of power envelopes for unit root tests under different
> conditions allowing for serial correlation, deterministic components,
> assumptions regarding the initial condition, non-Gaussian errors, and
> the use of covariates.

[Improving Size and Power in the Unit Root Testing](https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1147594)

Nevertheless, I can't help thinking that 16 annual observations is not much data to discipline time series estimation, and there will likely be lots of somewhat plausible specifications you will not be able to reject. Instead, you could use economy theory to justify the functional form and the statistical tests you use. For example, in efficient markets (crudely), price levels are highly serially correlated (random walks) but returns are independent. So if you knew your data were in returns, you might just argue rather than test that they are not a unit root.

> **Comments:**
> **username_0**: but is it okay if I will just use standard multiple ols regression for even if I have a small sample size?  
> **username_2**: That's a different question that should be asked separately.  

---

