An exchange rate model where the fundamentals follow a jump-diffusion process

Figures & data

Figure 1. Paths of the process X(t) with and without jumps.

Figure 2. Graph of the negative of the marginal log-likelihood evaluated as a mixture of 5, 10, and 30 components.

Figure 3. Histograms of the estimates of the parameters of the jump-diffusion process.

Table 1. Summary statistics

Variable	Minimum	Maximum	Mean	Median	Standard Deviation
Canada
$\mathrm{\Delta }f$ ^a	−0.0469	0.0578	−0.0023	−0.0030	0.0150
$\mathrm{\Delta }log(s)$ ^b	−0.0601	0.1129	0.0004	0.0002	0.0148
Japan
$\mathrm{\Delta }f$	−0.1770	0.0874	−0.0013	−0.0012	0.0215
$\mathrm{\Delta }log(s)$	−0.1052	0.0807	−0.0019	−0.0001	0.0267

^a$\mathrm{\Delta }f$ is the first difference of the fundamental series, $f$, which is derived as $f=m-m^{\ast }-(y-y^{\ast })$ (Mark, 1995), $m$ is the log of the US monetary aggregate, $m^{\ast }$ is the log of the corresponding foreign monetary aggregate, $y$ is the log of the US industrial production index, and $y^{\ast }$ is the log of the foreign industrial production index.

^b$\mathrm{\Delta }log(s)$ is the percentage change in the value of the foreign currency. $s$ is expressed in units of foreign currency per US dollar.

Table 2. Estimates of the parameters of the fundamental process for Canada and Japan

	ABM			ABM with jump
Parameters	estimates	ste^a	p-value	estimates	ste	p-value
Model for Canada
$\mu$	0.0100	0.0081	0.2194	−0.0333	0.0077	0.0000
$\sigma$ ^b	0.0531	0.0017	0.0000	0.0493	0.0017	0.0000
$k$				0.0437	0.0094	0.0000
$\lambda$ ^b				0.1440	0.0969	0.1374
log-likelihood	1531.6049			1547.9094
AIC	−3059.2099			−3087.8188
BIC	−3050.5719			−3070.5429
Model for Japan
$\mu$	0.0100	0.0111	0.3669	−0.0070	0.0099	0.4796
$\sigma$ ^b	0.0747	0.0023	0.0000	0.0673	0.0020	0.0000
$k$				−0.1220	0.0126	0.0000
$\lambda$ ^b				0.0667	0.0387	0.0847
log-likelihood	1342.2636			1380.8993
AIC	−2680.5271			−2753.7985
BIC	−2671.8892			−2736.5227

^aste = standard error

^bAll the p-values are two-sided, but $\sigma$ and $\lambda$ are restricted to be positive. Thus, appropriate p-value for these variables should be half the values in the table.

^c$AIC=-2\ast \mathrm{l}\mathrm{o}\mathrm{g}-\mathrm{l}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{d}+2\ast \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}$

^d$BIC=-2\ast \mathrm{l}\mathrm{o}\mathrm{g}-\mathrm{l}\mathrm{i}\mathrm{k}\mathrm{e}\mathrm{l}\mathrm{i}\mathrm{h}\mathrm{o}\mathrm{o}\mathrm{d}+log(n)\ast \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{s}$

Table 3. Estimates of $\alpha$ for Canada and Japan

	estimate	ste^a	p-value	J-stat	J-stat p-value
Canada	1162.9387	183.9531	6.3219	51.6241	0.0000
Japan	212.3611	5.7941	36.6513	557.0297	0.0000

^aste = standard error

Mark, N. (1995). Exchange rates and fundamentals: evidence on long-horizon predictability. American Economic Review, 85(1), 201–218. https://www.jstor.org/stable/2118004

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