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Asymmetry: Resurrecting the roots
The Quarterly Review of Economics and Finance 60 (2016) 201–206
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Asymmetry: Resurrecting the roots Manuel Frondel a,∗ , Christoph M. Schmidt a , Colin Vance b a b
Rheinisch-Westfälisches Institut für Wirtschaftsforschung and Ruhr University Bochum, Germany Rheinisch-Westfälisches Institut für Wirtschaftsforschung and Jacobs University Bremen, Germany
a r t i c l e
i n f o
Article history: Received 20 October 2014 Received in revised form 24 November 2014 Accepted 14 September 2015 Available online 9 October 2015 JEL classification: D13 Q41
a b s t r a c t This note provides for a didactic survey on a range of primary methods for dealing with price asymmetry. Using Wolffram’s (1971) stylized example, we argue that asymmetry can be captured in a straightforward and highly intuitive manner with first differences. While this asymmetry definition is more readily interpretable than the alternatives proposed by Wolffram (1971) and Houck (1977), we demonstrate that, theoretically, all three of these definitions are equivalent. Using data on U.S. coffee consumption, however, we illustrate that, in practice, these approaches may yield divergent conclusions on asymmetry. In such situations, the asymmetry concept based on first differences is advantageous. © 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.
Keywords: Irreversibility Decomposition approaches
1. Introduction The estimation of so-called irreversible supply and demand functions that allow for asymmetric price responses has been a subject of ongoing research across a range of fields, including agriculture (Mundlak & Larson, 1992; Rajcaniova & Pokrivcak, 2013; Traill, Colman, & Young, 1978) and energy economics (Adeyemi & Hunt, 2014; Bachmeier & Griffin, 2003; Cologni & Manera, 2009; Frondel & Vance, 2013; Griffin & Schulman, 2005; Peltzman, 2000). While theoretical arguments in favor of asymmetric responses to rising or falling agricultural input prices were advanced by Johnson (1958), the empirical work on the topic was pushed with an analysis of aggregate farm output by Tweeten and Quance (1969a, 1969b). Their approach, which employs dummy variables that split up the price variable into two complementing explanatory terms capturing either increasing or decreasing input prices, is criticized by Wolffram (1971: 356). Wolffram (1971) proposes an alternative technique based on cumulated price differences that, in their reply to his criticism, Tweeten and Quance (1971: 359) concede is superior to their approach, even though the application of the technique to their own data suggests otherwise (Tweeten & Quance, 1971: 360). In
∗ Corresponding author at: Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI), Hohenzollernstr. 1-3, D-45128 Essen, Germany. Tel.: +49 2018149204; fax: +49 2018149200.
the aftermath of this exchange, Wolffram’s technique, henceforth called the W technique, became the most popular method of partitioning an explanatory variable to allow for the estimation of a non-reversible function (Traill et al., 1978: 528), and has since served as a foundation for more sophisticated approaches, such as error-correction models (for helpful surveys, see Frey & Manera, 2007; Meyer & von Cramon-Taubadel, 2004). Despite Wolffram’s (1971) and Tweeten and Quance’s (1971) common belief of the superiority of the W technique, however, a number of articles have pointed to several weaknesses in its application, including the high dependence on the starting point of the data (Griffin & Schulman, 2005: 7) and its proneness to multi-collinearity problems (Saylor, 1974). Using Wolffram’s (1971) example originally conceived to demonstrate the superiority of his method over the Tweeten and Quance – henceforth TQ – approach, this note provides for a didactic survey on early asymmetry test approaches and argues that the notion of asymmetry can be captured in a straightforward and highly intuitive manner in terms of first differences. We prove that, in a deterministic context without stochastic influences, this asymmetry definition is equivalent to both Wolffram’s and Houck’s (1977) alternatives. Using an empirical example originating from the U.S. coffee market, however, we demonstrate that, in practice, these approaches may yield divergent conclusions with respect to asymmetry. We argue that in such situations the asymmetry concept based on first differences is advantageous for many reasons.
http://dx.doi.org/10.1016/j.qref.2015.09.002 1062-9769/© 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.
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Table 1 Wolffram’s original example and its modification. Original values
W technique
TQ technique
Modified y
y
x
a
y
x
w+
w−
w+
w−
x+
x−
x+
x−
y˜
20 35 29 44 59 44 35 70 90 84
10 13 11 14 17 12 9 16 20 18
−30 −30 −4 −26 −26 8 8 −10 −10 30
– 15 −6 15 15 −15 −9 35 20 −6
– 3 −2 3 3 −5 −3 7 4 −2
10 13 13 16 19 19 19 26 30 30
10 10 12 12 12 17 20 20 20 22
– 3 0 3 3 0 0 7 4 0
– 0 2 0 0 5 3 0 0 2
10 13 0 14 17 0 0 16 20 0
0 0 11 0 0 12 9 0 0 18
– 3 −13 14 3 −17 0 16 4 −20
– 0 11 −11 0 12 −3 −9 0 18
20 35 13 40 55 16 7 50 70 34
2. A reassessment of Wolffram’s example Wolffram (1971: 357) criticizes that any irreversible relationship y = f(x) between a dependent variable y and an explanatory variable x cannot be determined exactly with the TQ approach, which splits x into two complementary variables, x+ and x− . Variable x+ is defined as x1+ := x1 and for i > 1 by xi+
:= xi ,
if xi > xi−1 ,
(1)
xi+
and := 0 otherwise, where subscript i is used to denote the observation, while x− is defined in a similar way: x1− := 0, and for i > 1: xi− := xi ,
if xi ≤ xi−1 ,
(2)
and xi− := 0 otherwise. By definition, xi+ + xi− = xi for all i. As an alternative to the TQ decomposition of x, Wolffram (1971) suggests taking cumulated increases and decreases of the explanatory variable x, denoted here by wi+ and wi− , respectively. In detail, Wolffram (2000: 351–352) defines his approach by w1+ = w1− := x1 and for i > 1, + wi+ := wi−1 + Di+ · (xi − xi−1 ) = w1+ +
i
(xk − xk−1 )Dk+ ,
(3)
(xk − xk−1 )Dk− ,
(4)
k=2 − wi− := wi−1 − Di− · (xi − xi−1 ) = w1− −
i k=2
where Dk+ := 1 for xk > xk−1 and 0 otherwise, while Dk− := 1 − Dk+ .1 From this definition, it becomes obvious that w+ and w− include cumulated price in- and decreases, respectively. To demonstrate the superiority of his approach over the TQ decomposition, Wolffram (1971) conceives a straightforward example presented in Table 1. For this purpose, Wolffram (1971: 358) assumes the following exact relationship between the predefined values of dependent variable y and those of the explanatory variable x, which is split up into x+ and x− according to the TQ decomposition: yi = ai + 5xi+ + 3xi− .
(5)
In this equation, potential residual terms ui are set to zero: ui = 0, thereby attributing the varying differences between the predefined values yi and the predicted values yˆ i := 5xi+ + 3xi− to variable a, whose components are also shown in Table 1. As Wolffram (1971: 357) emphasizes, this contrasts with the classical Ordinary Least Squares (OLS) framework, in which variable a would adopt the role of a constant: a = a0 . It is not surprising,
1 Using the dummy variables Di+ and Di− , the TQ decomposition can be concisely described by xi+ = Di+ xi and xi− = Di− xi for i > 1 (Meyer & von Cramon-Taubadel, 2004: 594).
therefore, that when applying OLS methods, one obtains the following estimation equation for which both coefficient estimates, 6.25 and 6.99, differ greatly from the predefined coefficients in Eq. (5)2 : yi = −40.23(11.03) + 6.25(0.74)xi+ + 6.99(0.88)xi− + uˆ i ,
(6)
with R2 = 0.912, uˆ i = / 0 for all i, and standard errors reported in parentheses. In contrast, Wolffram shows that the correct coefficients 5 and 3 are reproduced – apart from the sign of coefficient 3 – by using the proposed W technique and regressing y on w+ and w− : yi = 0 + 5wi+ − 3wi− ,
(7)
where uˆ i = 0 for all i and, hence, R2 = 1. Obviously, this example was constructed in such a way that precisely this result will be obtained when using the W technique. In what follows, we demonstrate that Wolffram’s critique with regard to the TQ decomposition is generally correct, although it is inappropriate to blame the TQ decomposition for a poor performance in this specific example. The reason is that the differences between the coefficient estimates reported in Eq. (6) and the true coefficients of 5 and 3 is merely the result of the fact that the varying values ai are approximated by a constant when Eq. (5) is estimated by OLS. If one estimates Eq. (5) by employing variable a as an additional regressor, thereby avoiding any omitted-variable bias, one can exactly reproduce the coefficients given in Eq. (5). Furthermore, one point that immediately emerges from Wolffram’s example is that in case of irreversibility, one may expect distinct intercepts a+ and a− , a+ = / a− , as is shown in the following modification of Wolffram’s example: y˜ i = −30Di+ − 20Di− + 5xi+ + 3xi− , a+
−30Di+
a−
(8)
−20Di−
with = and = and the modified values y˜ i for the dependent variable being shown in Table 1. Eq. (8) reflects the fact that in case of asymmetry, one would expect two entirely distinct functions, one for each of the two different regimes of either increasing or decreasing values of x. If one falsely estimates Eq. (8) by using a common intercept, the following OLS results are obtained: y˜ i = −24.87(1.89) + 4.67(0.13)xi+ + 3.36(0.15)xi− .
(9) xi+
xi−
In statistical terms, the coefficient estimates of and are significantly different from the true vales 5 and 3, respectively. Clearly, these estimation results, which seem to support Wolffram’s criticism with respect to the TQ decomposition, are due to omittedvariable bias. This bias could be readily avoided by including two dummy variables that capture the different intercepts, rather than
2 Wolffram (1971: 358) reported an estimate of −43.16 for the constant, which appears to be wrong.
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In case of symmetry, that is, in the special case that H0 is true and, hence, ˇ : = ˇ+ = ˇ− , the relationship between y and x is also called reversible and (10) simplifies to yi = ˇxi (Di+ + Di− ) = ˇxi .
(11)
=1
By recursive iteration, the following equation for yi in levels can be derived from reversible relationship (11): yi = = = =
yi−1 + ˇ · (xi − xi−1 ) (for i > 1) yi−2 + ˇ · (xi−1 − xi−2 ) + ˇ · (xi − xi−1 ) (for i > 2) yi−2 − ˇ · xi−2 + ˇ · xi = . . . y1 − ˇ · x1 + ˇ · xi .
=0
In short, from reversible relationship (11) it follows that yi = ˇ · xi for all i ≥ 1. In a similar vein, a representation for yi can be gained from asymmetry definition (10) for i > 1: yi = yi−1 + ˇ+ · (xi − xi−1 )Di+ + ˇ− · (xi − xi−1 )Di− = ...
Fig. 1. Illustration of Wolffram’s example.
employing a common constant, thereby perfectly reproducing Eq. (8). Fig. 1 illustrates, however, that in the Wolffram example, the application of the TQ decomposition would require more than two distinct intercepts a+ and a− . Rather, while three individual intercepts are necessary to describe those parts of the graph with a positive slope of +5 by a linear function, three distinct intercepts are also required for the downward-sloping parts of the graph. From this illustration, it therefore follows that, generally, the TQ decomposition is not a viable procedure to capture asymmetric relationships, as in case of entirely different intercepts no degrees of freedom are left over. Furthermore, from this illustration a natural definition of asymmetry in terms of first differences xi : = xi − xi−1 and yi : = yi − yi−1 suggests itself: while for those parts of the graph with a positive slope the first differences of y and x are related by a factor ˇ+ = 5: yi = ˇ+ xi , the downward-sloping parts of the graph are linked by a factor ˇ− = 3: yi = ˇ− xi . (These proportions also become apparent from Table 1 and the respective columns related to the first differences of x and y.) Combining both the upwardand downward-sloping parts provides for a straightforward and highly intuitive definition of asymmetry that is well-known from the literature (Farrell, 1952; Houck, 1977): there is a (long-term) asymmetric relationship between two variables x and y if the null hypothesis H0 : ˇ+ = ˇ− can be rejected for the following equation of first differences (i > 1): yi = ˇ+ xi Di+ + ˇ− xi Di− .
(10)
In fact, apart from a constant ˇ that Houck (1977: 570) added to this specification, he started from asymmetry definition (10) to derive his approach to testing asymmetry, discussed in detail below. The empirical literature on asymmetric price responses commenced much earlier, though, with an article by Farrell (1952), who analyzes the demand Y for four habitual consumption goods (tobacco, beer, wine, and spirits) in response to variations in income (Z) and price (X). It deserves noting that the logarithmic transformation of Farrell’s estimation equation in levels takes a form that clearly reflects the asymmetry concept (10) based on first differences: yt = c + a+ zt Dt+ + a− zt Dt− + b+ xt Dt+ + b− xt Dt− + εt , where yt , xt , and zt denote logged demand Y, logged price X, and logged income Z (Frey & Manera, 2007: 358).
= y1 + ˇ+
i
i
(xk − xk−1 )Dk+ − ˇ− (−
k=2
=
k=2
=w− −w− 1 i + + − − + + − − y1 − (ˇ w1 − ˇ w1 ) + ˇ wi − ˇ wi .
=w+ −w+ 1 i
(xk − xk−1 )Dk− )
=0 if y1 =ˇ+ w+ −ˇ− w− 1
1
Hence, adopting asymmetry definition (10) implies that yi can be decomposed according to the W technique proposed by Wolffram (for comparison, see his example given by Eq. (7)): yi = ˇ+ wi+ − ˇ− wi−
for all i ≥ 1.
(12)
In short, asymmetry definitions (10) and (12) are equivalent in theory. Using OLS methods, this equivalence can also be easily confirmed for Wolffram’s empirical example presented in Table 1, for
ˇ+ = 5 and ˇ− = 3, while which one gets the following estimates: standard errors are vanishing for both coefficients due to the construction of an exact relationship between y and w+ and w− (see Eq. (7)). Houck (1977: 570) proposes an alternative approach that “is consistent with the Wolffram technique but is operationally clearer”. In fact, from a theoretical point of view, his approach is equivalent to Wolffram’s technique given by Eq. (12), as will be shown now. From Wolffram’s asymmetry approach (12) and, specifically, y1 = ˇ+ w1+ − ˇ− w1− , it follows for i > 1 that yi∗ := yi − y1 = ˇ+ (wi+ − w1+ ) − ˇ− (wi− − w1− ) = ˇ+ h+ − ˇ− h− , i i (13) where the explanatory variables (see definitions (3) and (4)): h+ := wi+ − w1+ = i
i
h+ i
and
h− i
are defined as follows
(xk − xk−1 )Dk+ ,
k=2 i
:= wi− − w1− = − h− i
(xk − xk−1 )Dk− .
(14)
k=2
In contrast to Wolffram’s specification (12), in Houck’s H approach both the dependent variable yi∗ := yi − y1 and the right-hand side of (13) are purged from any initial values. It bears noting that purging the right-hand side of (13) from the initial values could also be
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Table 2 Empirical comparison of asymmetry approaches. Tweeten, Quance y Coeffs. +
x x− w+ −w− h+ −h− xD+ xD− year Const. Adj. R2 Correlation Test: ˇ+ = ˇ− Coint. test: # Obs.: * **
−0.0052 −0.0072** – – – – – – −0.1566** 324.31** **
Wolffram y
Houck y*
First differences y
Errors
Coeffs.
Errors
Coeffs.
Errors
Coeffs.
Errors
(0.0013) (0.0018) – – – – – – (0.0137) (27.42)
– – −0.0072** −0.0013 – – – – 0.0477 −76.94
– – (0.0010) (0.0011) – – – – (0.0319) (62.29)
– – – – −0.0072** −0.0013 – – 0.0477 −94.87
– – – – (0.0010) (0.0011) – – (0.0319) (62.57)
– – – – – – −0.0042** −0.0015 0.0076 −15.12
– – – – – – (0.0012) (0.0013) (0.0054) (10.64)
0.743 (x+ , x− ) : −0.64 F(1, 48) = 2.76 Z(t) = −1.14 52
0.854 (w+ , w− ) : 0.97 F(1, 48) = 41.20** Z(t) = −2.64 52
0.854 (h+ , h− ) : 0.97 F(1, 48) = 41.20** Z(t) = −2.64 52
0.227 (xD+ , xD− ) : 0.18 F(1, 47) = 1.89 Z(t) = −8.55** 51
Statistical significance at the 5% level. Statistical significance at the 1% level.
Table 3 Asymmetry tests based on first-difference definition (13) and the error-correction models (15) and (16). Eq. (15)
ˇ1+ ˇ1− ˇ2+ ˇ2− ˇ1 ˇ2 ˇ3 ˇ4 + − Year Const. Adj. R2 Test: ˇ1+ = ˇ1− Test: ˇ2+ = ˇ2− Test: + = − Cointegr. test: # Observations:
Eq. (16)
Coeffs.
Errors
Coeffs.
Errors
Coeffs.
Errors
Coeffs.
Errors
−0.0021 −0.0039* −0.0022 −0.0023 – – – – −0.4272 0.2899 0.0095 −18.824
(0.0016) (0.0019) (0.0019) (0.0014) – – – – (0.3874) (0.3206) (0.0058) (11.469)
−0.0029* −0.0021 – – – – – – −0.6431* 0.1667 0.0071 −13.9795
(0.0014) (0.0016) – – – – – – (0.3236) (0.3284) (0.0055) (10.909)
– – – – −0.0028** – – – −0.7979** 0.2671 0.0069 −13.599
– – – – (0.0008) – – – (0.2915) (0.2381) (0.0053) (10.649)
– – – – −0.0024 −0.0016 −0.0021* −0.0002** −0.2162 0.0091 0.0106 −21.218
– – – – (0.0008) (0.0009) (0.0008) (0.0008) (0.3603) (0.3060) (0.0059) (11.692)
0.259 F(1, 41) = 0.37 F(1, 41) = 0.00 F(1, 41) = 1.32 Z(t) = −6.73** 49
0.270 F(1, 44) = 0.11 – F(1, 44) = 1.99 Z(t) = −6.96** 50
achieved by setting w1+ = w1− = 0, rather than w1+ = w1− = x1 , as is suggested by Wolffram (1971: 358). This consideration shows that the initial value x1 does not play any central role, a fact that is confirmed by our empirical estimations presented in the subsequent section. Actually, instead of Eq. (13), Houck (1977: 570) suggests a specification including a deterministic trend ˛t. This trend has been dropped here for the sake of simplicity, but included in the empirical example presented in the next section. Again, the equivalence of both Houck’s and Wolffram’s definitions can be confirmed for Wolffram’s example, for which the OLS estimates for the slope coefficients ˇ+ and ˇ− turn out to be the same, respectively, and standard errors vanish. In sum, while numerous approaches have been suggested in the economic literature to capture asymmetry, this section has demonstrated that, theoretically and for contrived examples, such as Wolffram’s, in which stochastic disturbances are absent, both Wolffram’s and Houck’s approaches are equivalent to asymmetry concept (10), which is based on first differences.3 However, for empirical examples originating from the real world, such as that presented in the subsequent section, we now demonstrate
0.302 – – F(1, 45) = 5.91* Z(t) = −6.86** 50
that Wolffram’s and Houck’s approaches and the definition based on first differences may yield contrary answers to the question of asymmetry. 3. Empirical illustration To illustrate this point, we present an empirical application that regresses per-capita coffee consumption in the U.S. on the price of robusta coffee beans. Coffee is a commodity that lends itself to investigation in the context of price asymmetries, as commodity price cycles lead to frequent and large price fluctuations. The
the asymmetry definition (10) based on first differences are all equivalent, Houck (1977) already proved that definition (10) is equivalent to his approach. This can be readily seen as follows: using yi∗ = yi − y1 =
i
k=2
(yk − yk−1 ) and definition (10) as
point of departure, Houck arrived at specification (13) as follows (i >1):
yi∗ = yi − y1 =
= ˇ 3
0.348 – – F(1, 39) = 0.39 Z(t) = −6.63** 47
+
i
k=2
While we have shown here that Houck’s and Wolffram’s definitions as well as .
i
(yk − yk−1 ) =
k=2
(xk −
xk−1 )Dk+
i
k=2
+ˇ
−
i
k=2
yk =
i
ˇ+ xi Di+ + ˇ− xi Di−
k=2
(xk − xk−1 )Dk− = ˇ+ h+ − ˇ− h− . i i
M. Frondel et al. / The Quarterly Review of Economics and Finance 60 (2016) 201–206
assembled data set is measured on a yearly basis spanning 1960 through 2011 and is compiled from two sources: the data on prices, which are expressed in real terms using the base year 2005, is taken from the Global Economic Monitor (GEM) Commodities web site of the World Bank (2013). The data on per-capita coffee consumption is drawn from the USDA Food Availability System. Referencing Eqs. (1)–(2) and (14), we transform the price series x using the TQ-, W and H decompositions, which serve as explanatory variables to explain per-capita consumption y. Furthermore, we add the variable year to account for secular trends in per-capita consumption. The empirical results obtained from the TQ- and W decompositions are compared in Table 2 to those received from the estimation of first-difference definition (10), as well as those from Houck’s approach. Several outcomes bear highlighting: First, apart from the constants, the empirical results of Wolffram’s and Houck’s specifications are identical. This is due to the fact that both the dependent variables yi and yi∗ and the key explanatory variables h+ , h− and i i wi+ , wi− differ merely by constants. In other words, Wolffram’s and Houck’s approaches are not only theoretically equivalent, as has been shown in the previous section, but are also identical from an empirical point of view. Second, while all key explanatory variables show the expected signs, yet are not always statistically significant, F tests clearly reject the null hypothesis of symmetry only for the Wolffram (= Houck) approach, but neither for the TQ specification nor for the approach based on first differences. This divergence raises the question as to which approach should be preferred when conclusions are drawn with respect to asymmetry, with the TQ decomposition of Tweeten and Quance (1969a, 1969b) being known to be an inferior option (Tweeten & Quance, 1971; Wolffram, 1971: 359). Using a modified version of Wolffram’s example (7) in which an error term is included that is drawn from a standard normal distribution, we first demonstrate that this issue cannot be resolved on the basis of statistical tests. In terms of estimation accuracy, for instance, it turns out that the first-difference approach is not superior to the W technique: While standard errors are lower for the W approach, the estimate 4.88 of ˇ+ = 5 resulting from yi = ˇ+ wi+ − ˇ− wi− = 4.88(0.09)wi+ − 2.87(0.11)wi− is closer to the true value 5 than the estimate stemming from the first-difference approach: yi = ˇ+ xi Di+ + ˇ− xi Di− = 4.84(0.16)xi Di+ + 2.99(0.23)xi Di− , whereas the opposite holds true for the estimates of ˇ− = 3. Nevertheless, we argue that, for at least four reasons, the asymmetry concept based on first differences bears advantages. First, while it is equivalent to Wolffram’s decomposition in a deterministic context, but is generally different in empirical examples with a limited number of observations,4 the basic principle of asymmetry is reflected in a highly transparent manner only by definition (10). Second, beyond this theoretical argument, the W technique is generally prone to multi-collinearity (Saylor, 1974). In the coffee example, for instance, the correlation coefficient between w+ and w− amounts to about 0.97, whereas the correlation between xD+ and xD− is substantially lower at 0.18. It bears noting, though, that in our example the large collinearity of w+ and w− does not
4
Using a simulation and a modification of Wolffram’s example that includes normally distributed error terms, we find indistinguishable coefficient estimates for both approaches for 10,000 observations, but substantially divergent estimates for only 100 observations. In this case, we also receive contradictory results for the issue of asymmetry.
205
inflate the standard errors and, hence, does not cause any empirical problems. Third, due to its formulation in first differences, definition (10) is also adequate in cases when the variables involved are I(1), as in our coffee example for which the Dickey–Fuller test statistics for x and y amount to Z(t) = −2.39 and Z(t) = −2.01, respectively. (The critical values for the 10%, 5%, and 1% significance levels are −2.600, −2.929, and −3.579, respectively.) In contrast, upon employing Dickey–Fuller tests, we can reject the null hypotheses that xD+ , xD− , and y are I(1): the test statistics read Z(t) = −4.72, Z(t) = −7.38, and Z(t) = −7.63, respectively. Relative to first-difference definition (10), the W approach may be more prone to spurious correlation, as our empirical example illustrates: Dickey–Fuller test statistics of Z(t) = −0.95 and Z(t) = −0.76 indicate that we cannot reject the null hypotheses that both variables, w+ and w− , are I(1). Furthermore, there is no co-integration relationship between y, w+ and w− , as the Dickey–Fuller test results reported in the penultimate row of Table 2 indicate. In fact, only for the first-difference approach (10) there exists a co-integration relationship, but not for the TQ, W, and H techniques. Therefore, fourth, employing Engle and Granger’s errorcorrection models as a more sophisticated method for simultaneously testing short- and long-term asymmetric behavior is in our coffee example merely possible for the first-difference approach (10), whereas error-correction models are not applicable to the TQ, W and H approaches due to the fact that the respective price variables and coffee demand are not co-integrated (Engle & Granger, 1987: 255–256). Following von Cramon-Taubadel and Loy (1996) and using our coffee example, we now test for asymmetry by estimating an error-correction model (ECM) that is based on the first-difference approach (i > 1):
yi = ˇ +
K
ˇj+ xi−j+1 Di+ +
j=1
+
+
ECT + i−1
L
ˇj− xi−j+1 Di−
j=1
+
−
ECT − i−1
+ εi ,
(15)
where for j > 1 xi−j+1 denotes lagged price changes and the lagged error-correction term ECT i−1 := yi−1 − ˇ+ xi−1 Di+ − ˇ− xi−1 Di− is split into two components: ECT + and ECT − , with ECT + = i−1 i−1 i−1 + ECT i−1 if ECTi−1 > 0 and ECT i−1 = 0 otherwise, whereas ECT − = i−1 5 ECT i−1 if ECTi−1 < 0 and ECT − = 0 otherwise. i−1 Table 3 indicates that the coefficients ˇ2+ and ˇ2− , which correspond to the lagged price difference xi−1 : = xi−1 − xi−2 , are not statistically significant, nor are those of higher lags of price differences (not reported) nor are the coefficients of the split lagged error correction terms, i.e. + and − . We thus have abstained from including any lagged price differences and have again estimated error-correction model (15), finding no asymmetry, neither in the short- nor in the long term. In fact, on the basis of the F test results reported in Table 3, we cannot reject the hypotheses H0 : + = − and H0 : ˇ1+ = ˇ1− individually, nor jointly (F test statistic: F(2, 41) = 0.77). Presuming hence a symmetric long-term relationship, that is, ˇ1+ = ˇ1− = ˇ1 , we have estimated the following special case of
5 Our definition of ECTi−1 deviates from that provided by Meyer and von CramonTaubadel (2004: 596), who define the lagged error-correction term as follows: ∗ ECT i−1 := yi−1 − ˇxi−1 . In the coffee example, though, y and x are not co-integrated, so that replacing ECTi−1 by ECT ∗i−1 in (15) would result in a non-stationary, rather than a co-integration relationship (15), as is reflected by the Dickey–Fuller test results reported in Table 3.
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error-correction model (15), see Meyer and von Cramon-Taubadel (2004: 596): yi = ˇ +
L
ˇj xi−j+1 + + ECT + + − ECT − + εi . i−1 i−1
(16)
j=1
When no lagged price differences are included, according to the F test result reported in Table 3, H0 : + = − can be rejected at the 5% significance level, implying that there is a short-term asymmetry from the long-term relationin that positive deviations ECT + i−1 ship are corrected more strongly than negative deviations ECT − . i−1 Estimation results for specifications in which multiple statistically significant lagged price differences are included indicate no such asymmetries, however (Table 3). In sum, the asymmetry tests based on both the first-difference definition and the estimation of errorcorrection models confirm the former outcome of no asymmetric demand responses due to coffee price changes (see Table 2). 4. Summary and conclusion Wolffram’s (1971) method has established itself as the point of departure for dealing with asymmetry within the field of agricultural economics, as well as other economic disciplines. This paper has demonstrated that Wolffram’s approach is principally consistent with an alternative definition of asymmetry that is based on first differences and highlighted here. While both approaches yield the same results for the stylized example given by Wolffram (1971), using an empirical example originating from the real world in which the data generation process is characterized by a stochastic component and the number of observations is limited, we have illustrated that both definitions may yield contrary answers to the question of asymmetry. This divergence raises the question as to which approach should be preferred when conclusions are drawn with respect to asymmetry. On the basis of our theoretical discussion, we argue that in such situations the definition of asymmetry based on first differences bears several advantages, not least because it is more easy to grasp than Wolffram’s W technique to capture asymmetry. With respect to prices, for instance, the W technique incorporates the history of the price trajectory by splitting up the price variable x into two complementary variables w+ and w− that reflect either cumulated price increases or decreases, respectively. This technique comes at some cost of intuition: because the W technique implies that the level of dependent variable y is supposed to be explained by cumulated changes of an explanatory variable x, it is not immediately clear how to interpret the coefficients. Beyond this, the W technique may be prone to spurious correlation, as our empirical coffee example has illustrated. In fact, as the first differences of time series variables are more likely to be integrated than the original series, the first-difference approach to testing asymmetry is less likely to be afflicted by spurious correlation than the W technique. Lastly, due to the absence of a co-integrated relationship between coffee demand and the price variables w+ and w− employed by the W technique, our empirical example has also demonstrated that applying more sophisticated
error-correction methods for testing asymmetry is only reasonable for the first-difference approach, yet not for the W technique. Acknowledgements This work has been supported by the Collaborative Research Center “Statistical Modeling of Nonlinear Dynamic Processes” (SFB 823) of the German Research Foundation (DFG), within the framework of Project A3, “Dynamic Technology Modeling”. References Adeyemi, O. L., & Hunt, L. C. (2014). Accounting for asymmetric price responses and underlying energy demand trends in OECD industrial energy demand. Energy Economics, 45, 435–444. Bachmeier, L., & Griffin, J. (2003). New evidence on asymmetric gasoline price responses. Review of Economics and Statistics, 85, 772–776. Cologni, A., & Manera, M. (2009). The asymmetric effects of oil shocks on output growth: A Markov–Switching analysis for the G-7 countries. Economic Modelling, 26, 1–29. Engle, R. F., & Granger, C. W. J. (1987). Co-integration and error correction: Representation, estimation and testing. Econometrica, 55(2), 251–276. Frey, G., & Manera, M. (2007). Econometric models of asymmetric price transmission. Journal of Economic Surveys, 21(2), 349–415. Farrell, M. J. (1952). Irreversible demand functions. Econometrica, 20(2), 171–186. Frondel, M., & Vance, C. (2013). Re-identifying the rebound: What about asymmetry? Energy Journal, 34(4), 43–54. Griffin, J. M., & Schulman, T. C. (2005). Price asymmetry: A proxy for energy saving technical change. Energy Journal, 26(2), 1–21. Houck, J. P. (1977). An approach to specifying and estimating nonreversible elasticities. American Journal of Agricultural Economics, 59, 570–572. Johnson, G. L. (1958). In E. O. Heady, H. G. Diesslin, H. R. Jensen, & G. L. Johnson (Eds.), Supply functions – Some facts and notions, in agricultural adjustments in a growing economy.. Ames: Iowa State University Press. Meyer, J., & von Cramon-Taubadel, S. (2004). Asymmetric price transmission: A survey. Journal of Agricultural Economics, 50, 581–611. Mundlak, Y., & Larson, D. F. (1992). On the transmission of world agricultural prices. World Bank Economic Review, 6(3), 399–422. Peltzman, S. (2000). Prices rise faster than they fall. Journal of Political Economy, 108, 466–502. Rajcaniova, M., & Pokrivcak, J. (2013). Asymmetry in price transmission mechanism: The case of Slovak potato market. Review of Agricultural and Applied Economics, 16(2), 16–23. Saylor, R. G. (1974). Alternative measures of supply elasticities: The case of Sao Paulo coffee. American Journal of Agricultural Economics, 56, 98–106. Traill, B., Colman, D., & Young, T. (1978). Estimating irreversible supply elasticities. American Journal of Agricultural Economics, 60, 528–531. Tweeten, L. G., & Quance, L. (1969a]). Positivistic measures of aggregate supply elasticities: Some new approaches. American Economic Review Papers and Proceedings, 59, 175–183. Tweeten, L. G., & Quance, L. (1969b]). Positivistic measures of aggregate supply elasticities: Some new approaches. American Journal of Agricultural Economics, 51, 342–352. Tweeten, L. G., & Quance, L. (1971). Techniques for segmenting independent variables in regression analysis: Reply. American Journal of Agricultural Economics, 53, 359–360. von Cramon-Taubadel, S., & Loy, J.-P. (1996). Price asymmetry in the international wheat market: Comment. Canadian Journal of Agricultural Economics, 44, 311–317. Wolffram, R. (1971). Positivistic measures of aggregate supply elasticities: Some new approaches – Some critical notes. American Journal of Agricultural Economics, 53, 356–359. Wolffram, R. (2000). Identifikation asymmetrischer Preisanpassungsprozesse für integrierte Zeitreihen – Eine Replik. Jahrbücher für Nationalökonomie und Statistik, 220(3), 350–357. World Bank. (2013). Global economic monitor (GEM) commodities, Washington. http://databank.worldbank.org/data/views/variableselection/selectvariables. aspx?source=global-economic-monitor-(gem)-commodities
Contents lists available at ScienceDirect
The Quarterly Review of Economics and Finance journal homepage: www.elsevier.com/locate/qref
Asymmetry: Resurrecting the roots Manuel Frondel a,∗ , Christoph M. Schmidt a , Colin Vance b a b
Rheinisch-Westfälisches Institut für Wirtschaftsforschung and Ruhr University Bochum, Germany Rheinisch-Westfälisches Institut für Wirtschaftsforschung and Jacobs University Bremen, Germany
a r t i c l e
i n f o
Article history: Received 20 October 2014 Received in revised form 24 November 2014 Accepted 14 September 2015 Available online 9 October 2015 JEL classification: D13 Q41
a b s t r a c t This note provides for a didactic survey on a range of primary methods for dealing with price asymmetry. Using Wolffram’s (1971) stylized example, we argue that asymmetry can be captured in a straightforward and highly intuitive manner with first differences. While this asymmetry definition is more readily interpretable than the alternatives proposed by Wolffram (1971) and Houck (1977), we demonstrate that, theoretically, all three of these definitions are equivalent. Using data on U.S. coffee consumption, however, we illustrate that, in practice, these approaches may yield divergent conclusions on asymmetry. In such situations, the asymmetry concept based on first differences is advantageous. © 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.
Keywords: Irreversibility Decomposition approaches
1. Introduction The estimation of so-called irreversible supply and demand functions that allow for asymmetric price responses has been a subject of ongoing research across a range of fields, including agriculture (Mundlak & Larson, 1992; Rajcaniova & Pokrivcak, 2013; Traill, Colman, & Young, 1978) and energy economics (Adeyemi & Hunt, 2014; Bachmeier & Griffin, 2003; Cologni & Manera, 2009; Frondel & Vance, 2013; Griffin & Schulman, 2005; Peltzman, 2000). While theoretical arguments in favor of asymmetric responses to rising or falling agricultural input prices were advanced by Johnson (1958), the empirical work on the topic was pushed with an analysis of aggregate farm output by Tweeten and Quance (1969a, 1969b). Their approach, which employs dummy variables that split up the price variable into two complementing explanatory terms capturing either increasing or decreasing input prices, is criticized by Wolffram (1971: 356). Wolffram (1971) proposes an alternative technique based on cumulated price differences that, in their reply to his criticism, Tweeten and Quance (1971: 359) concede is superior to their approach, even though the application of the technique to their own data suggests otherwise (Tweeten & Quance, 1971: 360). In
∗ Corresponding author at: Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI), Hohenzollernstr. 1-3, D-45128 Essen, Germany. Tel.: +49 2018149204; fax: +49 2018149200.
the aftermath of this exchange, Wolffram’s technique, henceforth called the W technique, became the most popular method of partitioning an explanatory variable to allow for the estimation of a non-reversible function (Traill et al., 1978: 528), and has since served as a foundation for more sophisticated approaches, such as error-correction models (for helpful surveys, see Frey & Manera, 2007; Meyer & von Cramon-Taubadel, 2004). Despite Wolffram’s (1971) and Tweeten and Quance’s (1971) common belief of the superiority of the W technique, however, a number of articles have pointed to several weaknesses in its application, including the high dependence on the starting point of the data (Griffin & Schulman, 2005: 7) and its proneness to multi-collinearity problems (Saylor, 1974). Using Wolffram’s (1971) example originally conceived to demonstrate the superiority of his method over the Tweeten and Quance – henceforth TQ – approach, this note provides for a didactic survey on early asymmetry test approaches and argues that the notion of asymmetry can be captured in a straightforward and highly intuitive manner in terms of first differences. We prove that, in a deterministic context without stochastic influences, this asymmetry definition is equivalent to both Wolffram’s and Houck’s (1977) alternatives. Using an empirical example originating from the U.S. coffee market, however, we demonstrate that, in practice, these approaches may yield divergent conclusions with respect to asymmetry. We argue that in such situations the asymmetry concept based on first differences is advantageous for many reasons.
http://dx.doi.org/10.1016/j.qref.2015.09.002 1062-9769/© 2015 The Board of Trustees of the University of Illinois. Published by Elsevier B.V. All rights reserved.
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Table 1 Wolffram’s original example and its modification. Original values
W technique
TQ technique
Modified y
y
x
a
y
x
w+
w−
w+
w−
x+
x−
x+
x−
y˜
20 35 29 44 59 44 35 70 90 84
10 13 11 14 17 12 9 16 20 18
−30 −30 −4 −26 −26 8 8 −10 −10 30
– 15 −6 15 15 −15 −9 35 20 −6
– 3 −2 3 3 −5 −3 7 4 −2
10 13 13 16 19 19 19 26 30 30
10 10 12 12 12 17 20 20 20 22
– 3 0 3 3 0 0 7 4 0
– 0 2 0 0 5 3 0 0 2
10 13 0 14 17 0 0 16 20 0
0 0 11 0 0 12 9 0 0 18
– 3 −13 14 3 −17 0 16 4 −20
– 0 11 −11 0 12 −3 −9 0 18
20 35 13 40 55 16 7 50 70 34
2. A reassessment of Wolffram’s example Wolffram (1971: 357) criticizes that any irreversible relationship y = f(x) between a dependent variable y and an explanatory variable x cannot be determined exactly with the TQ approach, which splits x into two complementary variables, x+ and x− . Variable x+ is defined as x1+ := x1 and for i > 1 by xi+
:= xi ,
if xi > xi−1 ,
(1)
xi+
and := 0 otherwise, where subscript i is used to denote the observation, while x− is defined in a similar way: x1− := 0, and for i > 1: xi− := xi ,
if xi ≤ xi−1 ,
(2)
and xi− := 0 otherwise. By definition, xi+ + xi− = xi for all i. As an alternative to the TQ decomposition of x, Wolffram (1971) suggests taking cumulated increases and decreases of the explanatory variable x, denoted here by wi+ and wi− , respectively. In detail, Wolffram (2000: 351–352) defines his approach by w1+ = w1− := x1 and for i > 1, + wi+ := wi−1 + Di+ · (xi − xi−1 ) = w1+ +
i
(xk − xk−1 )Dk+ ,
(3)
(xk − xk−1 )Dk− ,
(4)
k=2 − wi− := wi−1 − Di− · (xi − xi−1 ) = w1− −
i k=2
where Dk+ := 1 for xk > xk−1 and 0 otherwise, while Dk− := 1 − Dk+ .1 From this definition, it becomes obvious that w+ and w− include cumulated price in- and decreases, respectively. To demonstrate the superiority of his approach over the TQ decomposition, Wolffram (1971) conceives a straightforward example presented in Table 1. For this purpose, Wolffram (1971: 358) assumes the following exact relationship between the predefined values of dependent variable y and those of the explanatory variable x, which is split up into x+ and x− according to the TQ decomposition: yi = ai + 5xi+ + 3xi− .
(5)
In this equation, potential residual terms ui are set to zero: ui = 0, thereby attributing the varying differences between the predefined values yi and the predicted values yˆ i := 5xi+ + 3xi− to variable a, whose components are also shown in Table 1. As Wolffram (1971: 357) emphasizes, this contrasts with the classical Ordinary Least Squares (OLS) framework, in which variable a would adopt the role of a constant: a = a0 . It is not surprising,
1 Using the dummy variables Di+ and Di− , the TQ decomposition can be concisely described by xi+ = Di+ xi and xi− = Di− xi for i > 1 (Meyer & von Cramon-Taubadel, 2004: 594).
therefore, that when applying OLS methods, one obtains the following estimation equation for which both coefficient estimates, 6.25 and 6.99, differ greatly from the predefined coefficients in Eq. (5)2 : yi = −40.23(11.03) + 6.25(0.74)xi+ + 6.99(0.88)xi− + uˆ i ,
(6)
with R2 = 0.912, uˆ i = / 0 for all i, and standard errors reported in parentheses. In contrast, Wolffram shows that the correct coefficients 5 and 3 are reproduced – apart from the sign of coefficient 3 – by using the proposed W technique and regressing y on w+ and w− : yi = 0 + 5wi+ − 3wi− ,
(7)
where uˆ i = 0 for all i and, hence, R2 = 1. Obviously, this example was constructed in such a way that precisely this result will be obtained when using the W technique. In what follows, we demonstrate that Wolffram’s critique with regard to the TQ decomposition is generally correct, although it is inappropriate to blame the TQ decomposition for a poor performance in this specific example. The reason is that the differences between the coefficient estimates reported in Eq. (6) and the true coefficients of 5 and 3 is merely the result of the fact that the varying values ai are approximated by a constant when Eq. (5) is estimated by OLS. If one estimates Eq. (5) by employing variable a as an additional regressor, thereby avoiding any omitted-variable bias, one can exactly reproduce the coefficients given in Eq. (5). Furthermore, one point that immediately emerges from Wolffram’s example is that in case of irreversibility, one may expect distinct intercepts a+ and a− , a+ = / a− , as is shown in the following modification of Wolffram’s example: y˜ i = −30Di+ − 20Di− + 5xi+ + 3xi− , a+
−30Di+
a−
(8)
−20Di−
with = and = and the modified values y˜ i for the dependent variable being shown in Table 1. Eq. (8) reflects the fact that in case of asymmetry, one would expect two entirely distinct functions, one for each of the two different regimes of either increasing or decreasing values of x. If one falsely estimates Eq. (8) by using a common intercept, the following OLS results are obtained: y˜ i = −24.87(1.89) + 4.67(0.13)xi+ + 3.36(0.15)xi− .
(9) xi+
xi−
In statistical terms, the coefficient estimates of and are significantly different from the true vales 5 and 3, respectively. Clearly, these estimation results, which seem to support Wolffram’s criticism with respect to the TQ decomposition, are due to omittedvariable bias. This bias could be readily avoided by including two dummy variables that capture the different intercepts, rather than
2 Wolffram (1971: 358) reported an estimate of −43.16 for the constant, which appears to be wrong.
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203
In case of symmetry, that is, in the special case that H0 is true and, hence, ˇ : = ˇ+ = ˇ− , the relationship between y and x is also called reversible and (10) simplifies to yi = ˇxi (Di+ + Di− ) = ˇxi .
(11)
=1
By recursive iteration, the following equation for yi in levels can be derived from reversible relationship (11): yi = = = =
yi−1 + ˇ · (xi − xi−1 ) (for i > 1) yi−2 + ˇ · (xi−1 − xi−2 ) + ˇ · (xi − xi−1 ) (for i > 2) yi−2 − ˇ · xi−2 + ˇ · xi = . . . y1 − ˇ · x1 + ˇ · xi .
=0
In short, from reversible relationship (11) it follows that yi = ˇ · xi for all i ≥ 1. In a similar vein, a representation for yi can be gained from asymmetry definition (10) for i > 1: yi = yi−1 + ˇ+ · (xi − xi−1 )Di+ + ˇ− · (xi − xi−1 )Di− = ...
Fig. 1. Illustration of Wolffram’s example.
employing a common constant, thereby perfectly reproducing Eq. (8). Fig. 1 illustrates, however, that in the Wolffram example, the application of the TQ decomposition would require more than two distinct intercepts a+ and a− . Rather, while three individual intercepts are necessary to describe those parts of the graph with a positive slope of +5 by a linear function, three distinct intercepts are also required for the downward-sloping parts of the graph. From this illustration, it therefore follows that, generally, the TQ decomposition is not a viable procedure to capture asymmetric relationships, as in case of entirely different intercepts no degrees of freedom are left over. Furthermore, from this illustration a natural definition of asymmetry in terms of first differences xi : = xi − xi−1 and yi : = yi − yi−1 suggests itself: while for those parts of the graph with a positive slope the first differences of y and x are related by a factor ˇ+ = 5: yi = ˇ+ xi , the downward-sloping parts of the graph are linked by a factor ˇ− = 3: yi = ˇ− xi . (These proportions also become apparent from Table 1 and the respective columns related to the first differences of x and y.) Combining both the upwardand downward-sloping parts provides for a straightforward and highly intuitive definition of asymmetry that is well-known from the literature (Farrell, 1952; Houck, 1977): there is a (long-term) asymmetric relationship between two variables x and y if the null hypothesis H0 : ˇ+ = ˇ− can be rejected for the following equation of first differences (i > 1): yi = ˇ+ xi Di+ + ˇ− xi Di− .
(10)
In fact, apart from a constant ˇ that Houck (1977: 570) added to this specification, he started from asymmetry definition (10) to derive his approach to testing asymmetry, discussed in detail below. The empirical literature on asymmetric price responses commenced much earlier, though, with an article by Farrell (1952), who analyzes the demand Y for four habitual consumption goods (tobacco, beer, wine, and spirits) in response to variations in income (Z) and price (X). It deserves noting that the logarithmic transformation of Farrell’s estimation equation in levels takes a form that clearly reflects the asymmetry concept (10) based on first differences: yt = c + a+ zt Dt+ + a− zt Dt− + b+ xt Dt+ + b− xt Dt− + εt , where yt , xt , and zt denote logged demand Y, logged price X, and logged income Z (Frey & Manera, 2007: 358).
= y1 + ˇ+
i
i
(xk − xk−1 )Dk+ − ˇ− (−
k=2
=
k=2
=w− −w− 1 i + + − − + + − − y1 − (ˇ w1 − ˇ w1 ) + ˇ wi − ˇ wi .
=w+ −w+ 1 i
(xk − xk−1 )Dk− )
=0 if y1 =ˇ+ w+ −ˇ− w− 1
1
Hence, adopting asymmetry definition (10) implies that yi can be decomposed according to the W technique proposed by Wolffram (for comparison, see his example given by Eq. (7)): yi = ˇ+ wi+ − ˇ− wi−
for all i ≥ 1.
(12)
In short, asymmetry definitions (10) and (12) are equivalent in theory. Using OLS methods, this equivalence can also be easily confirmed for Wolffram’s empirical example presented in Table 1, for
ˇ+ = 5 and ˇ− = 3, while which one gets the following estimates: standard errors are vanishing for both coefficients due to the construction of an exact relationship between y and w+ and w− (see Eq. (7)). Houck (1977: 570) proposes an alternative approach that “is consistent with the Wolffram technique but is operationally clearer”. In fact, from a theoretical point of view, his approach is equivalent to Wolffram’s technique given by Eq. (12), as will be shown now. From Wolffram’s asymmetry approach (12) and, specifically, y1 = ˇ+ w1+ − ˇ− w1− , it follows for i > 1 that yi∗ := yi − y1 = ˇ+ (wi+ − w1+ ) − ˇ− (wi− − w1− ) = ˇ+ h+ − ˇ− h− , i i (13) where the explanatory variables (see definitions (3) and (4)): h+ := wi+ − w1+ = i
i
h+ i
and
h− i
are defined as follows
(xk − xk−1 )Dk+ ,
k=2 i
:= wi− − w1− = − h− i
(xk − xk−1 )Dk− .
(14)
k=2
In contrast to Wolffram’s specification (12), in Houck’s H approach both the dependent variable yi∗ := yi − y1 and the right-hand side of (13) are purged from any initial values. It bears noting that purging the right-hand side of (13) from the initial values could also be
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Table 2 Empirical comparison of asymmetry approaches. Tweeten, Quance y Coeffs. +
x x− w+ −w− h+ −h− xD+ xD− year Const. Adj. R2 Correlation Test: ˇ+ = ˇ− Coint. test: # Obs.: * **
−0.0052 −0.0072** – – – – – – −0.1566** 324.31** **
Wolffram y
Houck y*
First differences y
Errors
Coeffs.
Errors
Coeffs.
Errors
Coeffs.
Errors
(0.0013) (0.0018) – – – – – – (0.0137) (27.42)
– – −0.0072** −0.0013 – – – – 0.0477 −76.94
– – (0.0010) (0.0011) – – – – (0.0319) (62.29)
– – – – −0.0072** −0.0013 – – 0.0477 −94.87
– – – – (0.0010) (0.0011) – – (0.0319) (62.57)
– – – – – – −0.0042** −0.0015 0.0076 −15.12
– – – – – – (0.0012) (0.0013) (0.0054) (10.64)
0.743 (x+ , x− ) : −0.64 F(1, 48) = 2.76 Z(t) = −1.14 52
0.854 (w+ , w− ) : 0.97 F(1, 48) = 41.20** Z(t) = −2.64 52
0.854 (h+ , h− ) : 0.97 F(1, 48) = 41.20** Z(t) = −2.64 52
0.227 (xD+ , xD− ) : 0.18 F(1, 47) = 1.89 Z(t) = −8.55** 51
Statistical significance at the 5% level. Statistical significance at the 1% level.
Table 3 Asymmetry tests based on first-difference definition (13) and the error-correction models (15) and (16). Eq. (15)
ˇ1+ ˇ1− ˇ2+ ˇ2− ˇ1 ˇ2 ˇ3 ˇ4 + − Year Const. Adj. R2 Test: ˇ1+ = ˇ1− Test: ˇ2+ = ˇ2− Test: + = − Cointegr. test: # Observations:
Eq. (16)
Coeffs.
Errors
Coeffs.
Errors
Coeffs.
Errors
Coeffs.
Errors
−0.0021 −0.0039* −0.0022 −0.0023 – – – – −0.4272 0.2899 0.0095 −18.824
(0.0016) (0.0019) (0.0019) (0.0014) – – – – (0.3874) (0.3206) (0.0058) (11.469)
−0.0029* −0.0021 – – – – – – −0.6431* 0.1667 0.0071 −13.9795
(0.0014) (0.0016) – – – – – – (0.3236) (0.3284) (0.0055) (10.909)
– – – – −0.0028** – – – −0.7979** 0.2671 0.0069 −13.599
– – – – (0.0008) – – – (0.2915) (0.2381) (0.0053) (10.649)
– – – – −0.0024 −0.0016 −0.0021* −0.0002** −0.2162 0.0091 0.0106 −21.218
– – – – (0.0008) (0.0009) (0.0008) (0.0008) (0.3603) (0.3060) (0.0059) (11.692)
0.259 F(1, 41) = 0.37 F(1, 41) = 0.00 F(1, 41) = 1.32 Z(t) = −6.73** 49
0.270 F(1, 44) = 0.11 – F(1, 44) = 1.99 Z(t) = −6.96** 50
achieved by setting w1+ = w1− = 0, rather than w1+ = w1− = x1 , as is suggested by Wolffram (1971: 358). This consideration shows that the initial value x1 does not play any central role, a fact that is confirmed by our empirical estimations presented in the subsequent section. Actually, instead of Eq. (13), Houck (1977: 570) suggests a specification including a deterministic trend ˛t. This trend has been dropped here for the sake of simplicity, but included in the empirical example presented in the next section. Again, the equivalence of both Houck’s and Wolffram’s definitions can be confirmed for Wolffram’s example, for which the OLS estimates for the slope coefficients ˇ+ and ˇ− turn out to be the same, respectively, and standard errors vanish. In sum, while numerous approaches have been suggested in the economic literature to capture asymmetry, this section has demonstrated that, theoretically and for contrived examples, such as Wolffram’s, in which stochastic disturbances are absent, both Wolffram’s and Houck’s approaches are equivalent to asymmetry concept (10), which is based on first differences.3 However, for empirical examples originating from the real world, such as that presented in the subsequent section, we now demonstrate
0.302 – – F(1, 45) = 5.91* Z(t) = −6.86** 50
that Wolffram’s and Houck’s approaches and the definition based on first differences may yield contrary answers to the question of asymmetry. 3. Empirical illustration To illustrate this point, we present an empirical application that regresses per-capita coffee consumption in the U.S. on the price of robusta coffee beans. Coffee is a commodity that lends itself to investigation in the context of price asymmetries, as commodity price cycles lead to frequent and large price fluctuations. The
the asymmetry definition (10) based on first differences are all equivalent, Houck (1977) already proved that definition (10) is equivalent to his approach. This can be readily seen as follows: using yi∗ = yi − y1 =
i
k=2
(yk − yk−1 ) and definition (10) as
point of departure, Houck arrived at specification (13) as follows (i >1):
yi∗ = yi − y1 =
= ˇ 3
0.348 – – F(1, 39) = 0.39 Z(t) = −6.63** 47
+
i
k=2
While we have shown here that Houck’s and Wolffram’s definitions as well as .
i
(yk − yk−1 ) =
k=2
(xk −
xk−1 )Dk+
i
k=2
+ˇ
−
i
k=2
yk =
i
ˇ+ xi Di+ + ˇ− xi Di−
k=2
(xk − xk−1 )Dk− = ˇ+ h+ − ˇ− h− . i i
M. Frondel et al. / The Quarterly Review of Economics and Finance 60 (2016) 201–206
assembled data set is measured on a yearly basis spanning 1960 through 2011 and is compiled from two sources: the data on prices, which are expressed in real terms using the base year 2005, is taken from the Global Economic Monitor (GEM) Commodities web site of the World Bank (2013). The data on per-capita coffee consumption is drawn from the USDA Food Availability System. Referencing Eqs. (1)–(2) and (14), we transform the price series x using the TQ-, W and H decompositions, which serve as explanatory variables to explain per-capita consumption y. Furthermore, we add the variable year to account for secular trends in per-capita consumption. The empirical results obtained from the TQ- and W decompositions are compared in Table 2 to those received from the estimation of first-difference definition (10), as well as those from Houck’s approach. Several outcomes bear highlighting: First, apart from the constants, the empirical results of Wolffram’s and Houck’s specifications are identical. This is due to the fact that both the dependent variables yi and yi∗ and the key explanatory variables h+ , h− and i i wi+ , wi− differ merely by constants. In other words, Wolffram’s and Houck’s approaches are not only theoretically equivalent, as has been shown in the previous section, but are also identical from an empirical point of view. Second, while all key explanatory variables show the expected signs, yet are not always statistically significant, F tests clearly reject the null hypothesis of symmetry only for the Wolffram (= Houck) approach, but neither for the TQ specification nor for the approach based on first differences. This divergence raises the question as to which approach should be preferred when conclusions are drawn with respect to asymmetry, with the TQ decomposition of Tweeten and Quance (1969a, 1969b) being known to be an inferior option (Tweeten & Quance, 1971; Wolffram, 1971: 359). Using a modified version of Wolffram’s example (7) in which an error term is included that is drawn from a standard normal distribution, we first demonstrate that this issue cannot be resolved on the basis of statistical tests. In terms of estimation accuracy, for instance, it turns out that the first-difference approach is not superior to the W technique: While standard errors are lower for the W approach, the estimate 4.88 of ˇ+ = 5 resulting from yi = ˇ+ wi+ − ˇ− wi− = 4.88(0.09)wi+ − 2.87(0.11)wi− is closer to the true value 5 than the estimate stemming from the first-difference approach: yi = ˇ+ xi Di+ + ˇ− xi Di− = 4.84(0.16)xi Di+ + 2.99(0.23)xi Di− , whereas the opposite holds true for the estimates of ˇ− = 3. Nevertheless, we argue that, for at least four reasons, the asymmetry concept based on first differences bears advantages. First, while it is equivalent to Wolffram’s decomposition in a deterministic context, but is generally different in empirical examples with a limited number of observations,4 the basic principle of asymmetry is reflected in a highly transparent manner only by definition (10). Second, beyond this theoretical argument, the W technique is generally prone to multi-collinearity (Saylor, 1974). In the coffee example, for instance, the correlation coefficient between w+ and w− amounts to about 0.97, whereas the correlation between xD+ and xD− is substantially lower at 0.18. It bears noting, though, that in our example the large collinearity of w+ and w− does not
4
Using a simulation and a modification of Wolffram’s example that includes normally distributed error terms, we find indistinguishable coefficient estimates for both approaches for 10,000 observations, but substantially divergent estimates for only 100 observations. In this case, we also receive contradictory results for the issue of asymmetry.
205
inflate the standard errors and, hence, does not cause any empirical problems. Third, due to its formulation in first differences, definition (10) is also adequate in cases when the variables involved are I(1), as in our coffee example for which the Dickey–Fuller test statistics for x and y amount to Z(t) = −2.39 and Z(t) = −2.01, respectively. (The critical values for the 10%, 5%, and 1% significance levels are −2.600, −2.929, and −3.579, respectively.) In contrast, upon employing Dickey–Fuller tests, we can reject the null hypotheses that xD+ , xD− , and y are I(1): the test statistics read Z(t) = −4.72, Z(t) = −7.38, and Z(t) = −7.63, respectively. Relative to first-difference definition (10), the W approach may be more prone to spurious correlation, as our empirical example illustrates: Dickey–Fuller test statistics of Z(t) = −0.95 and Z(t) = −0.76 indicate that we cannot reject the null hypotheses that both variables, w+ and w− , are I(1). Furthermore, there is no co-integration relationship between y, w+ and w− , as the Dickey–Fuller test results reported in the penultimate row of Table 2 indicate. In fact, only for the first-difference approach (10) there exists a co-integration relationship, but not for the TQ, W, and H techniques. Therefore, fourth, employing Engle and Granger’s errorcorrection models as a more sophisticated method for simultaneously testing short- and long-term asymmetric behavior is in our coffee example merely possible for the first-difference approach (10), whereas error-correction models are not applicable to the TQ, W and H approaches due to the fact that the respective price variables and coffee demand are not co-integrated (Engle & Granger, 1987: 255–256). Following von Cramon-Taubadel and Loy (1996) and using our coffee example, we now test for asymmetry by estimating an error-correction model (ECM) that is based on the first-difference approach (i > 1):
yi = ˇ +
K
ˇj+ xi−j+1 Di+ +
j=1
+
+
ECT + i−1
L
ˇj− xi−j+1 Di−
j=1
+
−
ECT − i−1
+ εi ,
(15)
where for j > 1 xi−j+1 denotes lagged price changes and the lagged error-correction term ECT i−1 := yi−1 − ˇ+ xi−1 Di+ − ˇ− xi−1 Di− is split into two components: ECT + and ECT − , with ECT + = i−1 i−1 i−1 + ECT i−1 if ECTi−1 > 0 and ECT i−1 = 0 otherwise, whereas ECT − = i−1 5 ECT i−1 if ECTi−1 < 0 and ECT − = 0 otherwise. i−1 Table 3 indicates that the coefficients ˇ2+ and ˇ2− , which correspond to the lagged price difference xi−1 : = xi−1 − xi−2 , are not statistically significant, nor are those of higher lags of price differences (not reported) nor are the coefficients of the split lagged error correction terms, i.e. + and − . We thus have abstained from including any lagged price differences and have again estimated error-correction model (15), finding no asymmetry, neither in the short- nor in the long term. In fact, on the basis of the F test results reported in Table 3, we cannot reject the hypotheses H0 : + = − and H0 : ˇ1+ = ˇ1− individually, nor jointly (F test statistic: F(2, 41) = 0.77). Presuming hence a symmetric long-term relationship, that is, ˇ1+ = ˇ1− = ˇ1 , we have estimated the following special case of
5 Our definition of ECTi−1 deviates from that provided by Meyer and von CramonTaubadel (2004: 596), who define the lagged error-correction term as follows: ∗ ECT i−1 := yi−1 − ˇxi−1 . In the coffee example, though, y and x are not co-integrated, so that replacing ECTi−1 by ECT ∗i−1 in (15) would result in a non-stationary, rather than a co-integration relationship (15), as is reflected by the Dickey–Fuller test results reported in Table 3.
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M. Frondel et al. / The Quarterly Review of Economics and Finance 60 (2016) 201–206
error-correction model (15), see Meyer and von Cramon-Taubadel (2004: 596): yi = ˇ +
L
ˇj xi−j+1 + + ECT + + − ECT − + εi . i−1 i−1
(16)
j=1
When no lagged price differences are included, according to the F test result reported in Table 3, H0 : + = − can be rejected at the 5% significance level, implying that there is a short-term asymmetry from the long-term relationin that positive deviations ECT + i−1 ship are corrected more strongly than negative deviations ECT − . i−1 Estimation results for specifications in which multiple statistically significant lagged price differences are included indicate no such asymmetries, however (Table 3). In sum, the asymmetry tests based on both the first-difference definition and the estimation of errorcorrection models confirm the former outcome of no asymmetric demand responses due to coffee price changes (see Table 2). 4. Summary and conclusion Wolffram’s (1971) method has established itself as the point of departure for dealing with asymmetry within the field of agricultural economics, as well as other economic disciplines. This paper has demonstrated that Wolffram’s approach is principally consistent with an alternative definition of asymmetry that is based on first differences and highlighted here. While both approaches yield the same results for the stylized example given by Wolffram (1971), using an empirical example originating from the real world in which the data generation process is characterized by a stochastic component and the number of observations is limited, we have illustrated that both definitions may yield contrary answers to the question of asymmetry. This divergence raises the question as to which approach should be preferred when conclusions are drawn with respect to asymmetry. On the basis of our theoretical discussion, we argue that in such situations the definition of asymmetry based on first differences bears several advantages, not least because it is more easy to grasp than Wolffram’s W technique to capture asymmetry. With respect to prices, for instance, the W technique incorporates the history of the price trajectory by splitting up the price variable x into two complementary variables w+ and w− that reflect either cumulated price increases or decreases, respectively. This technique comes at some cost of intuition: because the W technique implies that the level of dependent variable y is supposed to be explained by cumulated changes of an explanatory variable x, it is not immediately clear how to interpret the coefficients. Beyond this, the W technique may be prone to spurious correlation, as our empirical coffee example has illustrated. In fact, as the first differences of time series variables are more likely to be integrated than the original series, the first-difference approach to testing asymmetry is less likely to be afflicted by spurious correlation than the W technique. Lastly, due to the absence of a co-integrated relationship between coffee demand and the price variables w+ and w− employed by the W technique, our empirical example has also demonstrated that applying more sophisticated
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