We propose a new variant of RESET that is appropriate for distributed lag models. Monte Carlo evidence on size and power strongly supports the use of the new variant instead of the traditional RESET.

Athanassios Stavrakoudis

Dimitris Hatzinikolaou

English

Research Papers in Economics

A New Variant of RESET for Distributed Lag Models 
Dimitris Hatzinikolaou Athanassios Stavrakoudis
University of Ioannina, Department of Economics University of Ioannina, Department of Economics
Abstract
We propose a new variant of RESET that is appropriate for distributed lag models. Monte
Carlo evidence on size and power strongly supports the use of the new variant instead of the
traditional RESET.
Citation: Hatzinikolaou, Dimitris and Athanassios Stavrakoudis, (2005) "A New Variant of RESET for Distributed Lag
Models." Economics Bulletin, Vol. 3, No. 56 pp. 1−4
Submitted: December 16, 2005.  Accepted: December 19, 2005.
URL: http://www.economicsbulletin.com/2005/volume3/EB−05C10023A.pdf  1
1. Introduction 
 
Ramsey’s (1969) regression specification error test (RESET) and its variants are known to 
have high power against certain alternatives, e.g., incorrect functional form, but low against 
others, e.g., omitted variables or omitted lags; see, e.g., Thursby (1989, Tables 5, 7, 11-13). 
In this paper, we propose a new variant of RESET, which has high power against omitted 
lags. Considering such a variant of RESET is important, because in empirical economics we 
often encounter distributed lag models, e.g., trade balance equations incorporating the J-curve 
effect, inflation equations, money market equations, etc., and the erroneous omission of lags 
from these models will in general lead to invalid statistical inference. Using both ordinary 
least squares (OLS) and the Cochrane-Orcutt (C-O) method, we produce Monte Carlo 
evidence on the size and power of the proposed variant as well as of Ramsey’s (1969) 
traditional RESET. We consider several true and null models and find that the proposed 
variant performs much better than the traditional RESET. After describing the test (Section 2) 
and our Monte Carlo setup (Section 3), we report our results and offer some possible 
explanations for the reported patterns (Section 4). Section 5 concludes the paper. 
 
2. The tests 
 
Consider the standard linear regression model,  v Xβ y + = , and assume that the data on y and 
X are stationary time-series. The RESET tests the hypothesis that this (null) model is 
specified correctly. Choose a T×M matrix Z of “test variables,” apply OLS to the equation 
u Zγ Xβ y + + = , (1) 
and test the hypothesis H0:  0 γ =  using a standard F test. Ramsey’s (1969) choice of test 
variables is  , ˆ (
2
t t Y = z  
3 ˆ
t Y , ...,  ) ˆ j
t Y , where  β x ˆ ˆ '
t t Y =  is the OLS fitted value from the null 
model. Let this test be denoted as POY(j). Our choice of test variables is 
2
1 ˆ ( − = t t Y z , 
2
2 ˆ
− t Y ,…, 
) ˆ 2
m t Y − , which is appropriate when testing a null distributed lag model for omitted lags, e.g., 
t i t i
k
i t v X Y + Σ + = − = β α 0  (2) 
against the alternative 
t i t i
l
i t u X Y + Σ + = − = β α 0 , (3) 
where l > k, m > k, and m may be greater than, equal to, or less than l. Let this variant of 
RESET be denoted as LOY(m). For example, if the researcher contemplates testing the null 
hypothesis of no lagged values of Xt (k = 0) against the alternative of l = 1 lag, then he/she 
can use LOY(1), which uses the test variable 
2
1 ˆ z − = t t Y ; if he/she contemplates testing k = 0 or 
k = 1 against l = 2 lags, then LOY(2) is appropriate, where 
2
1 ˆ ( − = t t Y z , 
2
2 ˆ
− t Y ); and so on. 
 
3. Monte Carlo design 
 
The data for Xt and ut are generated as follows: 
, 1 t t t X X ε ϕ + = −    , 95 . 0 , 5 . 0 , 0 . 0 = ϕ  (4)   
, 1 t t t w u u + = − ρ     , 95 . 0 , 5 . 0 , 0 . 0 = ρ  (5) 
t ε ∼ ) 10 , 5 ( . . . N d i i ,  t w ∼ ) 1 , 0 ( . . . N d i i , (6) 
0 X  ∼ )), 1 /( 10 ), 1 /( 5 (
2 ϕ ϕ − − N    0 u ∼ )) 1 /( 1 , 0 (
2 ρ − N . (7)   2
Then, using Equation (3), we generate data for Yt by assigning specific values to the 
parameters . ..., , , , , 1 0 l l β β β α  We consider six models, which emerge by assuming a 
maximum value of l = 3 and  , 5 . 0 , 6 . 0 , 0 . 10 2 1 0 = = = = β β β α  and  4 . 0 3 = β  in Equation (3). 
For each sample size, T = 50 and T = 200, and each combination of the parameters ϕ and 
ρ, we generate one set of T “observations” for the ε ’s and 5000 sets for the w’s, all from 
normal distributions, as indicated.
1 Then, we construct one set of “observations” for Xt, which 
we keep fixed in the 5000 replications; use Equation (3) to generate 5000 sets of T 
“observations” for Yt; and use them to estimate 5000 times the null model, Equation (2). In 
each experiment, we apply POY(j) and LOY(m) and calculate the proportion of rejections 
using a 5% level of significance. This proportion estimates the power of the test. We also 
estimate 5000 times the true model, Equation (3), apply POY(j) and LOY(m), and calculate 
the proportion of rejections, thus estimating the size of these tests. Note the following 
observations. 
First, in the presence of positive disturbance autocorrelation, POY(j) and LOY(m) tend to 
be oversized, especially when Xt is also positively autocorrelated, because the conventional 
standard errors are likely to underestimate the true ones, thus rendering the test variables 
spuriously significant; see, e.g., Johnston (1972, pp. 248-249) and Porter and Kashyap 
(1984). This can also occur because of a “spurious correlation” problem; see Leung and Yu 
(2001). Second, since we use a 5% nominal level of significance and the number of 
replications is 5000, the 95% confidence interval for the true percentage of rejections is (4.40, 
5.60). (This is a standard confidence interval for the population proportion, using a sample 
size of 5000 observations and a sample proportion of  . 05 . 0 ˆ = p ) Estimated sizes that fall 
outside this interval are regarded as significantly different from the nominal size. Third, we 
have generated POY(2), POY(3), and POY(4), but report the results for POY(2) only, 
because the three variants behave similarly, and because LOY(m) naturally compares to 
POY(2), since it uses second powers of lagged values of  t Y ˆ .  
 
4. Results 
 
Model 1: k = 0, l = 1. Using OLS and T = 50, we generate the rejection frequencies of 
POY(2) and LOY(1) and report them in Table 1: 
 
Table 1. Size and power of POY(2) and LOY(1) generated by OLS with T = 50 for Model 1 
   Size  Power 
ρ 
φ            
  0 0.50  0.95 0 0.50  0.95 
0 POY(2) 
LOY(1) 
5.46 
4.84 
5.24 
4.62 
5.44 
1.50
* 
1.76 
100.00 
0.90 
100.00 
1.02 
99.58 
0.50 POY(2) 
LOY(1) 
5.22 
5.08 
9.14
* 
7.16
* 
9.60
* 
23.88
* 
4.96 
100.00 
7.00 
100.00 
4.30 
95.82 
0.95 POY(2) 
LOY(1) 
5.26 
4.72 
19.02
* 
6.60
* 
31.58
* 
20.52
* 
0.08 
100.00 
2.86 
100.00 
15.32 
95.42 
Notes: (a) The rejection frequencies are given in percentages; (b) size estimates that fall outside the 
interval (4.40, 5.60) are regarded as significantly different from the nominal size (5%) and are marked 
by a star (*). 
                                                 
1 We use the LINUX version of RATS v. 5.01 to carry out the simulations. Random numbers were generated 
using the function %RAN(x) and the starting seed 317811. Note also that before we started drawing the values of 
ε and w, we let the process run for 500 “periods.”   3
Although the results on size are mixed, those on power clearly support the use of LOY(1). 
More precisely, the power of POY(2) is less than its size in most of our experiments, which 
suggests that POY(2) is a biased test for this application, whereas the power of LOY(1) 
always exceeds 95%. This is not surprising. In this case, Equation (1) is Yt = α + β0Xt + γZt + 
ut, where 
2 ˆ
t t Y Z =  for POY(2) and 
2
1 ˆ
− = t t Y Z  for LOY(1), and where  t Y ˆ  is the fitted value from 
the null model, Yt = α + β0Xt + vt. The variance estimator of the OLS coefficient γ ˆ is given by  
2 2
2
2
ˆ ) ( ) 1 ( Z Z r
S
S
t XZ − Σ −
= γ , (8) 
where S
2 is the residual variance from Equation (1) and  XZ r  is the correlation coefficient 
between Xt and Zt. Consider how the choice of Zt affects 
2
ˆ γ S . First, S
2 is expected to be larger 
when 
2 ˆ
t t Y Z =  than when 
2
1 ˆ
− = t t Y Z , since 
2
1 ˆ
− t Y  is a function of the omitted variable Xt-1, and 
should have more explanatory power in Equation (1) than does 
2 ˆ
t Y , which is a function of the 
already included variable Xt. The average values of S
2 from the 5000 replications confirm this 
expectation. Second, it seems hard to say a priori how the choice of Zt will affect 
2 ) ( Z Zt − Σ , but the average values of this sum from the 5000 replications are smaller when 
2 ˆ
t t Y Z =  than when 
2
1 ˆ
− = t t Y Z . Third, 
2
XZ r  is expected to be higher when 
2 ˆ
t t Y Z =  than when 
2
1 ˆ
− = t t Y Z , since 
2
0
2 ) ˆ ˆ ( ˆ
t t X Y β α + =  is highly correlated with Xt, whereas 
2
1 ˆ
− t Y  may not be 
correlated with Xt, unless ϕ  takes on a high value, e.g., ϕ = 0.95. The average values of 
2
XZ r  
are at least 0.9861 when 
2 ˆ
t t Y Z = , but range from 0.0155 to 0.9015 (increasing with ϕ) when 
2
1 ˆ
− = t t Y Z . For these three reasons, the standard error of γ ˆ when 
2 ˆ
t t Y Z =  is at least ten times 
greater than its counterpart when 
2
1 ˆ
− = t t Y Z , hence the big difference in empirical power.  
As was expected from our earlier discussion, the two tests are oversized when both φ ≥ 
0.50 and ρ ≥ 0.50. Thus, following Pagan and Hall (1983, pp. 206-209), we use the C-O 
method, which reduces the size distortion problem drastically: in only four (out of nine) 
experiments for each test the estimated size now falls outside the interval (4.40, 5.60), and 
then it only ranges from 4.14 to 7.28 for POY(2) and from 5.64 to 5.90 for LOY(1). Compare 
these ranges with those from OLS, 9.14 to 31.58 and 1.50 to 23.88, respectively. The C-O 
method also improves power: the power of POY(2) improves only at φ = 0, in which case it 
ranges from 36.66% to 55.28%, whereas that of LOY(1) is now 100% in every case. 
For space considerations, we will not report the results for T = 200. They differ from the 
case of T = 50 only in that (1) the power of LOY(1) is now 100% in every case, and (2) when 
the C-O method is used, only two size estimates fall outside the interval (4.40, 5.60), namely 
6.24 and 6.58 for POY(2) and 5.68 and 5.70 for LOY(1). 
Note that we have also applied LOY(2) and LOY(3). Compared to LOY(1), the only 
notable difference is that their size is generally greater, except when the C-O method is used 
with T = 200, in which case all size estimates fall within the interval (4.40, 5.60). 
Model 2: k = 0, l = 2. We use again POY(2) and LOY(1). Overall, considering both size 
and power, POY(2) behaves worse and LOY(1) behaves better than in Model 1. 
Model 3: k = 1, l = 2. In this case, we compare POY(2) with LOY(2). The conclusions are 
similar to those obtained in Models 1 and 2, when we compared POY(2) with LOY(1). It is 
worth noting that LOY(1) behaves better than LOY(2) in this case. 
Model  4:  k = 0, l = 3. Here, we compare POY(2) with LOY(1). The only notable 
difference from the previous cases is that when T = 50 and φ ≤ 0.50, the power of LOY(1)   4
generated by the C-O method ranges only from 71% to 89%. Note that LOY(2) is better in 
this case. 
Model 5: k = 1, l = 3. We use POY(2) and LOY(2). The results resemble those of Model 3. 
Model 6: k = 2, l = 3. We use POY(2) and LOY(3). The pattern of the results we have seen 
so far does not change. Note, however, that when T = 50, the power of LOY(3) generated by 
OLS is only 83% in two (out of nine) experiments. LOY(1) and LOY(2) are better. 
 
5. Conclusion 
 
This paper proposes a new variant of RESET that is appropriate for distributed lag models. 
Monte Carlo evidence on size and power suggests that the traditional RESET is a biased test 
in the present setup, whereas the new variant has good size properties, provided that it is 
generated by an autocorrelation robust method, and high power to detect the erroneous 
omission of lagged values of an explanatory variable.  
 
REFERENCES 
 
Johnston, J. (1972) Econometric methods, 2nd Ed., McGraw-Hill: Tokyo. 
Leung, S.F., and S. Yu (2001) “The sensitivity of the RESET tests to disturbance 
autocorrelation in regression residuals” Empirical Economics 26, 721-726. 
Pagan, A.R., and A.D. Hall (1983) “Diagnostic tests as residual analysis” Econometric 
Reviews 2, 159-218. 
Porter, R.D., and A.K. Kashyap (1984) “Autocorrelation and the sensitivity of RESET” 
Economics Letters 14, 229-233. 
Ramsey, J.B. (1969) “Tests for specification errors in classical linear least-squares regression 
analysis” Journal of the Royal Statistical Society, Series B 31, 350-371. 
Thursby, J.G. (1989) “A comparison of several specification error tests for a general 
alternative” International Economic Review 30, 217-230. 

A comparison of several specification error tests for a general alternative”

Autocorrelation and the sensitivity of

Diagnostic tests as residual analysis”

Econometric methods, 2nd Ed., McGraw-Hill:

Tests for specification errors in classical linear least-squares regression analysis”

A New Variant of RESET for Distributed Lag Models

http://www.accessecon.com/pubs/EB/2005/Volume3/EB-05C10023A.pdf

A New Variant of RESET for Distributed Lag Models

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