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This paper presents an empirical investigation of an important series called “economic fundamentals” derived from the flexible price monetary model of exchange rate determination. The model predicts that the nominal exchange rate is determined by the “economic fundamentals”, referred here as the series
f_{t}
. As a result, the characteristics of the “economic fundamentals” process may influence the properties of the nominal exchange rate process. We will just use the term “fundamentals”. Nevertheless, many exchange rate models found in the literature assume an
ad-hoc
process for
f_{t}
ignoring the fact that the specification of this process can be formally derived within the framework of the monetary model. Using data for several countries on GDP and money supplies, we construct the series
f_{t}
according to the mon
etary model specification, and we examine some important characteristics of its em
pirical distribution such as skewness, kurtosis, stationarity, ARCH and GARCH properties. We observe that the series is not exactly normally distributed, as commonly assumed in many target zone models. This investigation essentially helps with modeling exchange rate and more importantly in the analysis of exchange rate target zones modeling by identifying potential restrictions that need to be taken into consideration when choosing a process for the modeling of the “economic fundamentals”.

The monetary model was originally developed as a framework to analyze Balance-of- payments adjustments under a fixed exchange rate regime and has been modified after the Breakdown of the Bretton Woods system as a model of nominal exchange rate determination. The model is a common theme in textbooks on open-economy macroeconomics and in the literature. In addition, the model forms the basis of most target zone models and balance-of-payments studies. However, several versions of the monetary model have been developed over the years. In fact, Dornbusch [

The symbol

1) Money market equilibrium;

2) Continuous stock equilibrium in the money market;

3) Uncovered interest parity (UIP); and

4) Purchasing power parity.

Monetary equilibrium conditions in the domestic and foreign markets are given respectively by

The parameter

International capital market equilibrium is given by assuming that uncovered interest parity holds, that is,

where

Using the PPP condition, we obtain the fundamental equation of the flexible price monetary model

Under flexible exchange rates, the money stock is exogenous.

As commonly done in the literature, we simplify the model by assuming that the income elasticities and interest rate semi-elasticities of money demand are the same for the domestic and foreign countries. In this case, the fundamental equation becomes

At this point, we can use the UIP condition to obtain

Following Nelson Mark, we let

We therefore obtain

This last version constitutes the fundamental equation of the flexible price monetary model in discrete time. This equation is basically a first order stochastic difference equation in the variable

The expression for

To solve for

where

ly solved by the method of undetermined coefficients or by repeated substitutions. This equation states that expectations of future values of the exchange rate, that is,

It follows that

It follows that

In a similar fashion we obtain

Continuing the process, we obtain the following recursive equation

If the following transversality condition

is imposed, we obtain the so-called “no-bubbles” or “rational expectations” solution by letting k go to infinity:

The transversality condition puts a constraint on the rate at which the exchange rate can grow.

The “no-bubbles” solutions indicates that the exchange rate is the present value of expected future values of the economic fundamentals. This is in view with the “asset” approach to the exchange rate according to which the exchange rate should be expected to behave just like other assets, such as stocks and bonds.

We see that the exchange rate is a function of the fundamentals process. Therefore, the properties of the fundamentals have serious implications on the properties of the exchange rate process. Conversely, the observed characteristics of the exchange rate series put some restrictions on the potential characteristics of the fundamentals process. In this section we look at the implications of some well known stylized facts about exchange rate data for the characteristics of the fundamentals series.

For empirical purposes, we follow Mark characterization setting

Two common stylized facts are as follows:

1) The deviation of the price from the fundamentals displays substantial persistence and much less volatility than the exchange rate returns.

2) The volatility of exchange rate returns, that is of

At this stage of the analysis, we want to focus on some empirical analysis by specifically looking at some descriptive statistics about the economic fundamentals,

To address the excess volatility in the exchange rate returns relative to changes in the fundamentals, as in Mark [

where

The k-step ahead prediction formula is

Alternatively, we obtain

We then obtain

Thus, the exchange rate solution can be simplified and we have the following formula

Thus, we obtain the exchange rate solution to the standard FPMM

At this point, we want to compare the variance of the exchange rate returns,

Since the series

Hence, we see clearly that

Given the implications of the properties of the fundamentals for the characteristics of the exchange rate process, it is important to analyze some essential aspects of the empirical distribution of the fundamentals,

First, we present a set of sample statistics in

Japan | Sweden | UK | Canada | |
---|---|---|---|---|

Minimum | −0.2609 | −0.0596 | −0.1933 | −0.0126 |

Maximum | 1.0248 | 0.3175 | 2.0194 | 1.3654 |

Mean | 0.4276 | 0.1142 | 0.7214 | 0.7288 |

Median | 0.5140 | 0.1206 | 0.4540 | 0.8861 |

Std.Dev | 0.4196 | 0.0946 | 0.6519 | 0.4397 |

C.V. | 0.9813 | 0.8283 | 0.9037 | 0.6034 |

Skewness | −0.2323 | −0.0026 | 0.4138 | −0.2935 |

Ex. kurtosis | −1.3819 | −0.8627 | −1.3613 | −1.3769 |

nobs | 171 | 72 | 118 | 171 |

tribution. As a result, we will first look into the sample or empirical coefficient of kurtosis commonly defined as

It is clear that the distribution of the

A quick observation confirms our analysis, particularly for Japan and Sweden.

From this analysis, we observe that the series

Let

1)

2)

The t-statistics for the above tests as well as the corresponding p-value are presented in

According to the table at a 5% significance level we cannot reject the hypothesis that the distributions of the series are symmetric, but we can reject the hypothesis of zero excess kurtosis as well as the hypothesis of normality.

This analysis may seem unnecessary. However, it actually has very important implications in terms of exchange rate modeling in both the discrete and continuous time as well as aset pricing modeling. The following section tackles the issue of stationarity.

Given any time series, one important question that is often to be considered is whether

skew | p-value | kurt | p-value | JB | p-value | |||
---|---|---|---|---|---|---|---|---|

Japan | −0.2323 | −1.2403 | 0.2149 | 1.6181 | 4.3191 | 0.0000 | 20.1927 | 0.0000 |

Sweden | −0.0026 | −0.0089 | 0.9929 | 2.1373 | 3.7019 | 0.0002 | 13.7039 | 0.0011 |

UK | 0.4138 | 1.8349 | 0.0665 | 1.6387 | 3.6336 | 0.0003 | 16.5696 | 0.0003 |

Canada | −0.2935 | −1.5669 | 0.1171 | 1.6231 | 4.3325 | 0.0000 | 21.2260 | 0.0000 |

the series is stationary or non stationary. The examination of stationarity is important because doing regression analysis with non stationary time series can lead to spurious regression results. We now tackle this question for each of the

We conduct the analysis using the Augmented Dickey-Fuller test statistic or alternatively the KPSS test procedure. The first step is to identify the optimal order of the autoregressive process that fits the quarterly change in the fundamentals. To do so we start by plotting the sample autocorrelation function and the sample partial autocorrelation function for each series (f_t). For Japan we see that the first, the second, and the ninth partial autocorrelation coefficients appear to be significant at a 5% significance level suggesting an AR(2) or AR(9) model. Since the model that minimizes the Akaike Information criterion is an AR(9) and since the autocorrelation coefficients are significant for several lags, the autoregressive model of order 9 appears to be a good model for this series. A similar analysis for Sweden, the United Kingdom, and Canada suggest that we work with autoregressive processes of order 1, 2, and 3 respectively.

The Augmented Dickey Fuller test that follows is based on the above autoregressive models. The test’s null hypothesis assumes that the series is non stationary while the alternative hypothesis states that the series is stationary. Since the distribution of the ADF test statistic under the null hypothesis is not the usual t-distribution, Monte Carlo methods are used approximate the distribution of the statistic. The null hypothesis of nonstationarity is rejected as long as the test statistic is smaller than a commonly specified critical value or the associated p-value is less than a chosen significance level. Otherwise, stationarity should be maintained. In

Japan | Sweden | UK | Canada | |
---|---|---|---|---|

statistics | −2.8841 | −1.9489 | −0.7624 | −2.5585 |

Asympt. p-value | 0.2071 | 0.5961 | 0.9625 | 0.3431 |

Bootstrap p-value | 0.2229 | 0.6153 | 0.9527 | 0.3066 |

Lag order | 9.0000 | 1.0000 | 2.0000 | 3.0000 |

Japan | Sweden | UK | CANADA | |
---|---|---|---|---|

statistics | 2.6537 | 1.8265 | 3.6722 | 4.2132 |

p-value | 0.0100 | 0.0100 | 0.0100 | 0.0100 |

Lag order | 3.0000 | 1.0000 | 2.0000 | 3.0000 |

the low power of ADF test reported in the literature.

The KPSS test procedure is more commonly used in empirical work. The null hypothesis is that the series is stationary. Here, for completeness, we report both testing procedures.

To complete the analysis, we explore whether the series exhibit Autoregressive Conditional Heteroskedasticity (ARCH) or Generalized ARCH (GARCH) effects. Stock prices usually exhibit ARCH and GARCH properties. Moreover, given that most international economists believe that exchange rates behave like stock prices, we would expect exchange rates to have such properties. We will examine ARCH and GARCH effects in the next section.

We examine the conditional variance of the above four series to determine the presence of ARCH and GARCH effects. We start by estimating the conditional means of the series by determining the ARMA model that minimizes the Akaike Information criterion. We then generate the residuals which are used to compute the sample autocorrelation function and the sample partial autocorrelation functions of the squared residuals. They are plotted in the following figures (

According to the sample autocorrelation functions of the square of the residuals, Japan and the UK cannot be modeled as a pure Moving Average process. However, a Moving Average process of order 0 seems to be appropriate for Sweden, and in the case of Canada one may be able to use a Moving Average process of order 3.

Further, looking at the sample partial autocorrelation functions, one can see that the first, and the ninth lags are significant for Japan, which suggests a possible AR(1) mo-

del; for the UK an AR(1) model may be appropriate; the series for Sweden seem to be white noise, while for Canada the partial autocorrelation function seems to point toward an AR(3). The above observation suggests an ARCH(1) for Japan, an ARCH(1) for the UK, a GARCH(3,3) for Canada, and no ARCH effect for Sweden.

We also use the Akaike information criterion (AIC) to identify the best autoregressive models for the square of the residuals. The models obtained are ARCH(1) for Japan and the UK and ARCH(0) for Sweden and Canada.

To test for ARCH effects or the presence of ARCH effects, the most commonly used test is the Lagrange multiplier(LM) test proposed by Engle. The test is usually performed by first estimating the best fitting regression equation usually called the “mean equation”, which can be an appropriate ARMA or AR model and then use the residuals to estimate the coefficients. For example, to test for an ARCH(2) model for Japan, we collect the squared residuals from the autoregressive model with the smallest AIC (AR(9)) and then run the regression

Of course,

As the table indicated there is evidence in favor of conditional heteroskedasticity for Japan and the UK, but we cannot reject the hypothesis that residuals for Sweden and Canada are white noise.

The estimates of the corresponding ARCH models for Japan and the UK are presented in

LM statistic | p value | ARCH order (q) | |
---|---|---|---|

Japan | 26.90 | 0.00 | 2.00 |

Sweden | 0.97 | 0.33 | 1.00 |

UK | 22.25 | 0.00 | 2.00 |

Canada | 1.82 | 0.18 | 1.00 |

Japan | UK | |||||||
---|---|---|---|---|---|---|---|---|

Estimate | Std. Error | t value | Estimate | Std. Error | t value | |||

mu | −0.00 | 0.00 | −2.71 | 0.01 | −0.00 | 0.00 | −1.79 | 0.07 |

ar1 | 0.21 | 0.09 | 2.33 | 0.02 | 0.55 | 0.09 | 6.18 | 0.00 |

ar2 | −0.00 | 0.10 | −0.03 | 0.98 | 0.22 | 0.08 | 2.59 | 0.01 |

ar3 | 0.02 | 0.12 | 0.18 | 0.86 | ||||

ar4 | 0.04 | 0.05 | 0.90 | 0.37 | ||||

ar5 | 0.09 | 0.06 | 1.51 | 0.13 | ||||

ar6 | 0.17 | 0.06 | 3.00 | 0.00 | ||||

ar7 | −0.01 | 0.06 | −0.20 | 0.84 | ||||

ar8 | −0.09 | 0.09 | −1.05 | 0.29 | ||||

ar9 | 0.13 | 0.06 | 2.25 | 0.02 | ||||

omega | 0.00 | 0.00 | 2.44 | 0.01 | 0.00 | 0.00 | 5.24 | 0.00 |

alpha1 | 0.29 | 0.14 | 2.13 | 0.03 | 0.53 | 0.17 | 3.11 | 0.00 |

alpha2 | 1.00 | 0.33 | 3.04 | 0.00 |

In this paper, we have analyzed some empirical aspects of the economic fundamentals that is shown to be the driving force of exchange rate behavior. The analysis shows that this fundamental series does not appear to be normally distributed as indicated by the excess kurtosis and skewness as well as the estimated kernel density as indicated by the data for several countries. Also, we observe that some ARCH effects are present for most data while GARCH effects are not that common. This has important implications for modeling exchange rate and also the analysis of target zone models.

Cupidon, J.R. and Hyppolite, J. (2016) An Empirical Investigation of the Monetary Model Economic Fun- damentals. Modern Economy, 7, 1728-1740. http://dx.doi.org/10.4236/me.2016.714151