Seasonal fluctuations and equilibrium models of exchange rate

Abstract

Most of the evidence on dynamic equilibrium exchange rate models is based on seasonally adjusted consumption data. Equilibrium models have not worked well in explaining the actual exchange rate. However, the use of seasonally adjusted data might be responsible for the spurious rejection of the model. This article presents a new equilibrium model for the exchange rates that incorporates seasonal preferences. The fit of the model to the data is evaluated for five industrialized countries using seasonally unadjusted data. Our findings indicate that a model with seasonal preferences can generate monthly time series of the exchange rate without seasonality even when the variables that theoretically determine the exchange rate show clear seasonal behaviours. Further, the model can generate theoretical exchange rates with the same order of integration than actual exchange rates, and in some cases, with the same stochastic trend.

Notes

¹There is a three-member representative household, each of whom goes his own way during a period, the three regrouping at the end of a day to pool goods, assets, and information: the owner of the firm collects the production; a second member of the household takes an amount of the household's initial cash balances and uses it to purchase goods from other households; and the third member carries out the remaining domestic and foreign currency cash balances on the securities and foreign exchange markets where domestic and foreign securities and currencies are sold and bought.

²The attributes of the value function -existence, differentiability, and so on- are the subject of a vast technical literature. For main results and references, the interested reader should delve into Stockey and Lucas (1989).

³The amount of commodity that can be purchased with a unit of j currency at time t + 1 is the inverse of its price . The utility from each one of these units of is given by its marginal utility,

⁴For instance, for the UK pound/US dollar exchange rate case: the marginal utility expected from the investment of one pound on UK shares is the marginal utility expected from the investment of one pound on US shares is that is, the US-dollar value of a pound (1/S_t ) multiplied by the marginal utility expected from the investment on US shares.

⁵We assume that the utility function is additively separable from the two consumption services and presents constant intertemporal substitution elasticity. Although this assumption is restrictive, it simplifies estimation considerably. Our analysis focuses on a simple basic utility function with seasonal shocks. We think this is valuable for setting a benchmark. Even so, the literature on equilibrium models of exchange rates includes heterogeneity of agents, multiple sectors, tax shocks and modifications designed to reproduce features of exchange rates. Whether our findings are relevant for these cases is an open, quantitative issue that may be addressed using the procedures implemented in this article.

⁶This solution is the perfectly pooled equilibrium of Lucas (1982). Cole and Obstfeld (1991) show that the efficient allocations of resources in an economy like the one considered in this article is of the form: It is simple to see that when these consumption expressions are assumed, proportionality factors in the consumption rules cancel out, leading exactly to the same expression (26). That means that, so long as countries are following efficient consumption allocations, the exchange rate will be determined by the same variables.

⁷To gain some intuition, let us focus on one season in which is relatively low (if γ^j  > 1), then will be relatively high during this season, unless the seasonal parameter be relatively high to avoid seasonal fluctuations in the marginal utility of expected returns from investment. The taste shifts parameters indicate that the utility obtained from the consumption at time t varies according the prevailing season S and this explain that m ^j _t+1 does not track seasonal patterns in . Agents improve their welfare avoiding the creation of intertemporal distortions when making their decision to invest.

⁸At date t, domestic and foreign agents know past and present stock prices and dividends, good prices, outputs, and money endowments: for j = D, F. All investors observe the same signals and thus share information set. To understand what this information set entails consider an economy that exists over time periods t = 1, 2, … Let X_t denote the information set available to agents at date t. Let denote a possible choice vector for an agent in period t. Let X denote the collection of all possible agent information sets, and let H denote the collection of all possible choice vectors. Then a sequence (h ₁, h ₂, …) of agent choice vectors is said to be chosen in accordance with a policy rule if there exists some well-defined function g: X → H mapping information sets into choice vectors such that: h_t  = g(X_t ), t = 1, 2, …. The solution to the domestic agent's optimization problem is a feasible policy rule g*: X → H that determines how a particular policy vector h_t should be chosen from X on each period t ≥ 1 given the dynamic state Equations 10 and 11 that reveal how a choice of h_t in period t affects all current an future states of the economy.

⁹The utility function assumed in (23) is additively separable over time and additively separable in domestic and foreign goods. This assumption allows us to estimate seasonal taste parameters, factor discount and intertemporal elasticity of substitution parameters using single equation methods for each good. This can be seen by noting that in Equation 32, u_{t +1} is a function only of the variables and asset returns corresponding to country j (= D, F).

¹⁰We use Heteroskedasticity and an Autocorrelation Consistent Covariance Matrix of the sample moments.

¹¹Hansen (1982) showed that sufficient conditions for the asymptotic properties of the GMM include strict stationarity of the data. Strict stationarity may be violated for some kinds of seasonal variation. However, consistency and asymptotic normality of the estimators and the asymptotic distribution of the test statistic can be demonstrated under weaker conditions. See Jagannathan (1983) and Lim (1985) for an analysis of the asymptotic properties of the GMM under seasonality and nonstationarity. Stationarity may also be violated under some models of growth rates of real outputs or monetary aggregates. In our empirical study, these growth rates are stationary.

¹²Kydland and Prescott (1982) found that they needed a value of between one and two to mimic the observed relative variability of consumption and investment. Kehoe (1983), studying the response of small countries’ balance of trade to terms of trade shocks, obtains estimates near one. Mehra and Prescott (1985) present evidence for restricting the value of relative risk aversion to a maximum of 10. Hansen and Singleton (1982) report values of γ between zero and one. Mankiw (1985) reports values of between 2 and 4.

¹³Note that preferences given by expressions (23) and (24) assume that seasonal taste coefficients vary across domestic and foreign goods. Risk aversion for each good is also different. So the finding of different and γ ^j for goods produced in different countries does not contradict the assumption of identical preferences.

¹⁴The currency is calculated as the value of the second country's currency. For example, (GBP/USD) is the number of British pounds needed to purchase a US dollar; in this case, the UK is the domestic country and the USA the foreign country.

¹⁵Equation 35, E[f(w_{t +1}, z_t , )] = 0, represents a set of population orthogonality conditions from which an estimator of the preference parameters () may be obtained. Following Hansen and Singleton (1982), we use as instrumental variables (z_t ) p lags of the variables included in w_t+1  ≡ (, (μ_{t +1}(s) − μ_t (s))). Therefore, parameter estimation is based only on T-p observations. Because of the seasonal component of some of these variables – e.g., production growth rates, monetary aggregates growth rates – , long length (up to 48 months) of lagged values of w_{t +1} are needed. In order to show a homogeneous sample analysis, the theoretical exchange rate simulation is started at date 1990:01.

¹⁶Empirical results do not show any increased ability of the model to explain exchange rate volatility. Volatility of the simulated exchange rate series is an order of magnitude smaller that the volatility of the actual exchange rate series. The analysis including nontraded goods or assuming market segmentation would increase the capacity of this model to replicate high volatility in the exchange rates, but to obtain closed-form solutions for the exchange rate and make the model amenable to intuition and empirical research, we assume that prices are flexible and PPP holds at every point. The article must be viewed as a first attempt to incorporate seasonal preferences into a flexible price equilibrium model of exchange rate. While it may not have resolved all the equilibrium model problems, the development of more generalized preference structure allow us to replicate some stochastic properties of the actual exchange rate for some currencies: (1) no seasonal fluctuation and (2) degree of integration equal to one.

¹⁷Both Engle and Granger and Johansen cointegration tests reject the null of no-cointegration between theoretical and observed variables for the same pair of exchange rates: GDP/USD and ESP/DEM. Later evidence is provided when error-correction models (ECM) are estimated for these two pairs of exchange rates. Augmented Dickey Fuller t-ratio test and Z-test suggested in Phillips (1987) (see Phillips and Ouliaris, 1990) failed to reject the null of no-cointegration for both pair of estimated cointegration vectors. The Lagrange-multiplier statistic for first-order serial correlation in residuals of the ECM shows that they do not suffer from serial correlation and the Lagrange-multiplier statistic for first order autoregressive heteroskedasticity suggests that there is not evidence of autoregressive conditional heteroskedasticity. Normality of residuals is tested by the Jarque–Bera statistic. We only found deviations from normality in the residuals of the ECM associated with the observed ESP/DEM exchange rate. This long-tailed distribution is explained by two outliers detected in 10/92 and 5/93 due to the European Monetary System crisis. However, an intervention analysis reveals that cointegration tests were not distorted.