CC BY
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DOI https://doi.org/10.18551/rjoas.2017-07.02
THE EFFECTS OF OUTLIERS ON THE TEST FOR SYMMETRY IN PRICE
TRANSMISSION MODELS
Acquah De-Graft H., Associate Professor Department of Agricultural Economics and Extension, University of Cape Coast,
Cape Coast, Ghana E-mail: [email protected]
ABSTRACT
Monte Carlo experimentation is conducted to investigate the effects of outlier observations on the test of symmetry in the Houck and Granger and Lee asymmetric price transmission data generating process. The results of the Monte Carlo experimentation indicate that fewer outliers show little modification in the probability of Type I error rates for the test of symmetry in the Houck and Granger and Lee models. However, rejection frequencies of the true null hypothesis of symmetry rises with increase in the number of outliers in the data generating process. The simulation results indicate that the severity of outlier effects on the test of symmetry depends on sample size and the underlying asymmetric price transmission data generating process. With large amount of outliers and large sample size, the Houck's model detects more spurious asymmetries in the symmetric data generating process than the Granger and Lee model. These results confirm the claims of Douglas (2010) that outliers in the price data could generate asymmetric price response.
KEY WORDS
Outliers, Monte Carlo simulations, power, asymmetric price transmission models, type I error.
An important area of research within price transmission analysis has focused on investigating asymmetries in the price transmission process. Subsequently, numerous studies (Douglas (2010), Meyer and von Cramon-Taubadel (2004), Abdulai (2002), Peltzman (2000), von Cramon-Taubadel (1998), Borenstein et al. (1997), Hahn (1990), Bailey and Brorsen (1989), Kinnucan and Forker (1987) and Ward (1982) find evidence of price asymmetry in agricultural and related markets. However, the evidence of price asymmetry or non-linearity could result from the presence of outliers in data. Kind (2015) notes that price data used in price transmission analysis contains outliers. This is worrying since measuring asymmetry with data containing outliers could lead to misleading inference on price asymmetry and policy implications. The importance of outliers in asymmetric price transmission analysis is noted in previous research using empirical price data. For example, Douglas (2010) claims that asymmetry found is caused by outliers in data. He notes that asymmetry disappears after exclusion of outliers in the data.
The effect of outliers on hypothesis testing has been mentioned several times in the economics and statistics literature. In the economics literature some studies (Ahmad & Glosser 2011; Koop & Potter, 2001; Van Dijk, Franses & Lucas, 1999; Ahmed, 2008; Lopez Villavicencio, 2008; Ahamed & Donayre, 2002; and Tolvi, 2000) examined the effect of outliers on the test for non-linearity. These studies found that non-linearity is reflected in a small number of outlier observations in the data that they examined. They noted that outlier observations can distort linear relationship and cause the linearity test to reject the correct null hypothesis of linearity too often. However, these studies did not investigated the effect of outliers on commonly used asymmetric price transmission models (Houck and Granger and Lee models). In the statistics literature, some studies (Liao, Li & Brooks, 2016; Barnett & Lewis, 1994; Hampel, Ronchetti, Rousseeuw, & Stahel, 1986; Wilcox, 1998; and Zimmerman, 1994b) have provided evidence that shows the effect of outliers resulted in inflation of Type I error rates and reduced power in parametric t and F tests.
Aside Douglas (2010) claims that outliers cause price asymmetry, little is known about the effect of outliers on the test for non-linearity in asymmetric price transmission analysis.
Empirically, less effort has been made in examining the effect of outliers on the test for non-linearity within the asymmetric price transmission modelling context. Notably, the ability of the commonly used asymmetric price transmission models to cause the linearity test to reject the correct null hypothesis of linearity too often in the presence of outliers have not yet been extensively investigated and is not well understood. An important question which remains unanswered is how well will the test for non-linearity perform in the Houck and Granger and Lee models when outliers are present in the data used for price transmission analysis. In the presence of outliers, will there be inflation of Type I error rates and reduced power in the test for non-linearity in asymmetric price transmission models? And will the severity of outlier effects on the test of symmetry depend on sample size and the underlying asymmetric price transmission data generating process?
In order to address this issue, this paper supports Douglas (2010)'s claim that outliers in price data causes asymmetry and empirically evaluate the effect of outliers on the test for symmetry in two commonly used asymmetric price transmission models ( Houck and Granger and Lee models) via Monte Carlo experimentation. The paper contributes towards understanding the effect of outliers on the test for symmetry in Houck and Granger and Lee Models. The true data generating process is known in all experiments and the Monte Carlo simulations are essential in deriving the Type I error rates. Specifically, using simulated data, it is shown that the presence of outliers in a symmetric price data generating process lead to spurious asymmetries and could make the symmetry test reject the true null hypothesis of symmetry.
First, the study evaluates the power of the test for symmetry in the symmetric Houck's and Granger and Lee data generating process when outliers are absent. Second, the effect of outliers on the test for symmetry is investigated by introducing outliers into the symmetric Houck's and Granger and Lee data generating process. This study differs from existing literature in that it is the only one that explores the effect of outliers on the Houck and Granger and Lee asymmetric price transmission data generating process in a Monte Carlo experimentation.
METHODS OF RESEARCH
Measuring Asymmetry. Wolffram (1971) proposed an approach to measuring asymmetric price transmission that draws from Tweenten and Quance (1969). Houck (1977) suggested a simple but operationally clear alternative to the Wolffram approach to measuring asymmetric price transmission. However, these approaches do not take into consideration the long-run equilibrium relationship between the variables. Von Cramon Taubadel (1998) provides an alternative approach to measuring asymmetry that takes into consideration the concept of cointegration which considers the long- run equilibrium relationship between the variables. This model draws from Granger and Lee (1989).
Measuring Asymmetry using the Houck's Model. Houck (1977) suggested a simple model for testing for asymmetry as follows:
Ay t = Ax+ + Ax; + s s~ N(0,a2E) (1),
Where: Ax+ and Axt are the positive and negative changes in xt. The explanatory variable xt is generated as independent draws from normal distribution with a constant mean and a variance of one. In order to introduce asymmetry into the transmission process, speeds of adjustments for the coefficients of Ax+ and Axt in eqn. (1) are allowed to differ and the errors (s) are generated as independently and identically distributed draws from the standard normal distribution with a sample size n. Ayt can be constructed using the values
for beta, positive and negative changes in xt (i.e. Ax+ and Ax;) and the error term as
specified in eqn. (1). Symmetric price transmission behavior is tested by determining whether the coefficients (p and p ) are identical (i.e.H0 : Pj+ = P\_ ).
Measuring Asymmetry using Granger and Lee Model. Consider the following simple Granger and Lee (1989) asymmetric error correction model data generating process which shows the adjustment in variable y in response to deviations from its equilibrium relationship with variable x. Notably, yt and xt are generated as non-stationary variables that are
integrated of the order one. A cointegrating relationship exist between y and x which is defined by the error correction term. The positive and negative components of the error correction term are denoted by (yt - xt)+t-1 and (yt - xt )"t_.
The errors s are normally distributed with a mean of zero and a variance of one.
Ay = QxAxt _ 6>2+ (y _ x)+_\ _ 6>2_(y _ x)-_! + st (2)
To allow for asymmetric adjustments, the equilibrium relationship is partitioned as follows:
+ =| (y_x)t-\if (y_x)t-i >0
t-1 I 0 Otherwise (3)
( y _ x)+_\ ={
+
( y _ Y ) " — ( (y " x )i-iif (y " x )i-i <0
yy Jt -1_ |0 Otherwise (4)
Symmetry behavior in eqn. (2) is detected by determining whether the coefficients (d and d ) are identical (that is H0: d2+ — d2_).
RESULTS AND DISCUSSION
In order to investigate the effect of outliers on the test for asymmetry under various sample size conditions, a series of Monte Carlo simulations of the Houck and Granger and Lee models are carried out based on 10000 replications. In particular, effect of various percentages of outliers on the test for non-linearity in two asymmetric price transmission models (Houck and Granger and Lee models) are investigated under conditions of different sample sizes when the true data generating process is symmetric with the levels of asymmetry given by:
(p2+, p 2 _) — (d2+ ) e (0.25,0.25)
The Houck's symmetric data generating process is specified as follows:
A y t — 0.25 Axt+ + 0.25 A x_ + s (5)
The Granger and Lee symmetric data generating process is specified as follows: Ayt — 0.50Axt - 0.25(y - x)^ - 0.25(y - x)_-1 + s (6)
The assignment of asymmetric adjustment parameters draws from Holly et. al., (2003). The errors are generated from a normal distribution with a mean 0 and a variance of 1 for data without outliers. Data with outliers is created by introducing various percentages of outliers (0-5 percents) into the data. For example, one percent of the number of observations
of the errors generated for the normal data with values generated from a normal distribution with a mean of 0 and a variance of 1, were replaced with one percent of the number of observations from the normal distribution with a mean of 20 and variance of 1 for a chosen sample size. This is repeated for 2, 3, 4 and 5 percent of outliers given the various sample sizes 150, 300 and 500 respectively. The data generating processes are simulated 10000 times with different percentages of outliers and across different sample sizes. The performance of the symmetry test of the Houck and Granger and Lee models are evaluated in terms of their ability to reject the true null hypothesis of symmetric adjustment using an F-test of the restricted versus the unrestricted model when the data has no outliers or is contaminated with outliers. The results in Table 1 indicate the rejection frequencies of the true null hypothesis of symmetry. For an example, a rejection frequency of 0.0506 in Table 1 for 0 outliers and a sample size of 150 imply that the null hypothesis of symmetry is rejected 5.06% of the time for the Granger and Lee model.
Table 1 - Rejection Frequencies for Symmetry Test - Granger and Lee Model
Outlier % Sample Size(N) and 5% Rejection Frequency
N Rej (5%) N Rej (5%) N Rej (5%)
0 150 0.0506 300 0.0540 500 0.0518
1 150 0.0501 300 0.0500 500 0.0500
2 150 0.0528 300 0.0970 500 0.3827
3 150 0.0537 300 0.3203 500 0.8003
4 150 0.0909 300 0.6128 500 0.9655
5 150 0.2510 300 0.9613 500 0.9976
Based on 10,000 replications.
The study examined the rejection frequencies of the true null hypothesis of symmetry when there are no outliers in the simulated data. Notably, the study finds asymmetry in its absence. The results show that the Type I error rates of the F-test was around the significance level of 0.05 when the data has no outliers as illustrated in upper part of Tables 1 and 2. Similarly, Liao, Li and Brooks (2016) found that when outliers are removed from the dataset, the Type I error rates of ANOVA tests drop back to a significance level of around 0.05.
Table 2 - Rejection Frequencies for Symmetry Test - Houck's Model
Outlier % Sample Size (N) and 5% Rejection Frequency
N Rej (5%) N Rej (5%) N Rej (5%)
0 150 0.0506 300 0.0500 500 0.0500
1 150 0.0501 300 0.0515 500 0.0500
2 150 0.0509 300 0.0999 500 0.3830
3 150 0.0527 300 0.3230 500 1.0000
4 150 0.0950 300 0.6024 500 1.0000
5 150 0.2375 300 0.9629 500 1.0000
Based on 10,000 replications.
The study also examined the rejection frequencies of the true null hypothesis of symmetry when the data is contaminated with outliers. The study found a clear effect of outliers on the test for non-linearity. Fewer outliers show little modification in the probability of Type I error rates for the test of symmetry. This becomes evident when rejection frequencies obtained for data with 0 and 1 percent outlier contamination are compared under the Houck and Granger and Lee models in Tables 1 and 2. Similarly, Liao, Li and Brooks (2016) note that fewer outliers show little modification in the probability of Type I error rates. However, increasing the number of outliers in the various samples increased the probability of the Type I error for the test of symmetry. Increasing the outlier percentage from 2% to 5% increased the rejection frequencies of the true null hypothesis of symmetry for different sample sizes. For example, in Table 1, as outlier percentages increase from 2% to 5%, rejection
frequencies of the true null hypothesis of symmetry in the Granger and Lee model increased from 5.28% to 25.10%, 9.70% to 96.13% and 38.27% to 99.76% for sample sizes of 150, 300 and 500 respectively at 5% significance levels. Similarly, in Table 2 as outlier percentages increase from 2% to 5%, rejection frequencies of the true null hypothesis of symmetry in the Houck's model increased from 5.09% to 23.75%, 9.99% to 96.29% and 38.30% to 100% for sample sizes of 150, 300 and 500 respectively at 5% significance level.
Table 3 - Rejection Frequencies for Symmetry Test - Granger and Lee Model
Outlier % Sample Size(N) and 1% Rejection Frequency
N Rej (1%) N Rej (1%) N Rej (1%)
0 150 0.0101 300 0.0115 500 0.0113
1 150 0.0117 300 0.0108 500 0.0103
2 150 0.0126 300 0.0206 500 0.1291
3 150 0.0144 300 0.0957 500 0.4559
4 150 0.0198 300 0.2558 500 0.7815
5 150 0.0632 300 0.7563 500 0.9467
Based on 10,000 replications.
Table 4 - Rejection Frequencies for Symmetry Test - Houck's Model
Outlier % Sample Size(N) and 1% Rejection Frequency
N Rej (1%) N Rej (1%) N Rej (1%)
0 150 0.0101 300 0.0109 500 0.0109
1 150 0.0110 300 0.0112 500 0.0104
2 150 0.0103 300 0.0236 500 0.1270
3 150 0.0123 300 0.0958 500 0.9970
4 150 0.0190 300 0.2605 500 0.9996
5 150 0.0603 300 0.7630 500 1.0000
Based on 10,000 replications.
Similar patterns are observed for the Houck and Granger and Lee models at 1% significance levels in Table 3 and 4. Notably, as outlier percentages increase from 2% to 5%, rejection frequencies of the true null hypothesis of symmetry in the Granger and Lee model increased from 1.26% to 6.32%, 2.06% to 75.63% and 12.91% to 94.67% for sample sizes of 150, 300 and 500 respectively at 1% significance level in Table 3. Similarly, in Table 4 as outlier percentages increase from 2% to 5%, rejection frequencies of the true null hypothesis of symmetry in the Houck's model increased from 1.03% to 6.03%, 2.36% to 76.30% and 12.70% to 100% for sample sizes of 150, 300 and 500 respectively at 1% significance level. In effect rejection frequencies of the true null hypothesis of symmetry rises as the amount of outliers increase in individual samples. Similarly, some studies (Barnett & Lewis, 1994; Hampel, Ronchetti, Rousseeuw, & Stahel, 1986; Wilcox, 1998; Zimmerman, 1994b) have provided evidence that shows the effect of outliers resulted in inflation of Type I error rates and reduced power in parametric t and F tests. Similarly, Liao, Li and Brooks (2016) also found that with an increasing number of outliers being injected into the sample, the probability of the Type I error rates of the ANOVA test increased substantially.
Other factors which influences the rejection frequencies of the true null hypothesis of symmetry (Type I error rates) are the sample size and underlying asymmetric data generating process. At 5% outlier contamination, there is some increase in rejection frequencies when the sample size is increased 150 to 500 for both the Houck and the Granger and Lee models. At 5% outlier contamination, rejection frequencies for the true null hypothesis of symmetry in the Granger and Lee model are 25.10%, 96.13% and 99.76% for sample sizes of 150, 300 and 500 respectively at 5% significance level. Similarly, at 5% outlier contamination, rejection frequencies for the null hypothesis of symmetry in the Houck's model are 23.75%, 96.29% and 100% for sample sizes of 150, 300 and 500 respectively at 5% significance level. These results suggest that as sample size increases, the Type I error rates increases but at a decreasing rate. Using simulated populations, Liao, Li and Brooks (2016) noted that the magnitude of Type I error rate inflation decreases with
the growth of sample size. In other words, the impact of outliers on the false rejection rates is substantially greater with smaller sample sizes, and as sample size increases, the impact of outliers decreases although it is still inflated. With large amount of outliers (5%) and large sample size, the Houck's model detect more spurious asymmetries in the symmetric data generating process than the Granger and Lee model. These results indicate that the severity of outlier effects on the test of symmetry depends on sample size as well as the underlying asymmetric data generating process.
CONCLUSION
The study examined the effect of outliers on the test for symmetry in the Houck and Granger and Lee asymmetric price transmission models. Generally, the Monte Carlo simulation results indicate that fewer outliers show little change in the probability of Type I error rates for the test of symmetry in the Houck and Granger and Lee models. However, increasing the amount of outliers increase the ability of the symmetry test to reject the correct null hypothesis of symmetry in the asymmetric price transmission models studied. In summary, the presence of outliers lead to spurious asymmetries and could make the symmetry test reject the correct null hypothesis of symmetry. These results confirm the claims of Douglas (2010) that outliers in the price data could generate price asymmetry. With large amount of outliers (5%) and large sample size, the Houck's model detect more spurious asymmetries in the symmetric data generating process than the Granger and Lee model. In effect, simulation results indicate that the severity of outlier effects on the test of symmetry depends on sample size and the underlying asymmetric price transmission data generating process.
The study contributes to knowledge and understanding of the effects of outliers on the test for symmetry in asymmetric price transmission modelling framework. Additionally, the study contributes to the literature on asymmetric price transmission modelling by making researchers aware of the tendency of the test for symmetry to reject correct null hypothesis of symmetry in the asymmetric price transmission models when outlier percentages in data are high.
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