Method | RDP 2025-09: Forecasts of Period-average Exchange Rates: Insights from Real-time Daily Data

RDP 2025-09: Forecasts of Period-average Exchange Rates: Insights from Real-time Daily Data 4. Method

Martin McCarthy and Stephen Snudden

December 2025

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4.1 Out-of-sample evaluation

We conduct an out-of-sample evaluation of forecasts of the month-average exchange rate. Although exchange rates are observed for all countries, our baseline sample uses 83 countries for which all types of exchange rates (bilateral NER, bilateral RER, NEER, REER) start no later than 1 January 1994. Using a common sample period and set of countries facilitates comparisons between the results for different types of exchange rates. For each of these countries, we produce real-time forecasts using each monthly vintage. To ensure that all forecasts are made with models estimated on at least 10 years of data, the forecast evaluation sample uses monthly vintages from January 2004 to September 2022.

When computing forecast errors for bilateral RERs, we target the actual outcome computed from the bilateral NERs and CPI data as at end June 2023. For EERs, we target the actual outcome computed with the bilateral NERs and CPI levels as at end June 2023, but with the weights known on the forecast date. In this case, the forecaster needs to predict the combined effect of changes in bilateral NERs and CPI levels, but not the weights. This approach best reflects the aims of policymakers, who typically do not try to predict the effect of future changes in weights, in part because new weights will typically not be released until after the end of the forecast horizon. This approach also ensures that the treatment of trade weights in the later forecast vintages are consistent with the earlier forecast vintages.

The sample of 83 countries includes those with various exchange arrangements (floating, fixed, other managed arrangements), including countries whose exchange arrangements changed part way through the sample period (e.g. Lithuania, whose currency was pegged to the USD, then pegged to the euro, and then replaced by the euro). We include all countries in the forecast exercise and treat them equally. In doing so, we do not attempt to account for structural breaks such as changes in exchange rate regimes. We do this because we aim to quantify the effects of temporal aggregation generally rather than to take a stand on the best forecast practices for any specific country.

We employ two common real-time forecast evaluation criteria.

The first forecast evaluation criteria is the ratio of the root mean square forecast error (RMSFE) of a candidate model relative to the RMSFE of the benchmark. Specifically, the RMSFE ratio at horizon h, $R M S F E_{h}^{r a t i o}$ , is computed as the quotient of the RMSFE of the model-based forecast and the RMSFE of the alternative forecast:

(3)

R M S F E_{h}^{r a t i o} = \sqrt{\frac{\frac{1}{M} {\sum_{m = 1}^{M} (A_{m + h} - {\hat{A}}_{m + h | m}^{c a n d i d a t e})}^{2}}{\frac{1}{M} {\sum_{m = 1}^{M} (A_{m + h} - {\hat{A}}_{m + h | m}^{b e n c h})}^{2}}}

where ${\hat{A}}_{m + h | m}^{c a n d i d a t e}$ represents the real-time candidate forecast for the h step ahead of forecast target $A_{m + h}$ , and ${\hat{A}}_{m + h | m}^{b e n c h}$ is the alternative benchmark forecast, for all periods of the evaluation sample, denoted as m = 1,...,M. We also perform Diebold-Mariano tests (Diebold and Mariano 1995) of the null that expected squared error loss is equal. To perform the test for a horizon h, we compute a loss differential for forecasts at that horizon (i.e. difference in squared errors). We then regress the loss differentials on an intercept, and use Newey and West (1987) standard errors. The two-sided test of the null that the intercept is zero uses standard normal critical values.

The second forecast criteria assesses directional accuracy and is computed using the success ratio (SR). The SR describes the fraction of times the forecasting model can correctly predict the change in direction of the series of interest relative to the benchmark:

(4)

S R_{h} = \frac{1}{M} \sum_{m = 1}^{M} 𝟙 [sgn (A_{m + h} - {\hat{A}}_{m + h | m}^{b e n c h}) = sgn ({\hat{A}}_{m + h | m}^{c a n d i d a t e} - {\hat{A}}_{m + h | m}^{b e n c h})]

where sgn(.) is a sign function and $𝟙$ [.] is an indicator function taking the value of one if true and zero otherwise. We also test the null of no directional accuracy by testing if the categorical random variables $sgn (A_{m + h} - {\hat{A}}_{m + h | m}^{b e n c h})$ and $sgn ({\hat{A}}_{m + h | m}^{c a n d i d a t e} - {\hat{A}}_{m + h | m}^{b e n c h})$ are independent of each other. The test statistic is calculated following Pesaran and Timmermann (2009).

4.2 Description of forecasting methods

This subsection describes the forecasting methods. These methods are in three broad categories: no change forecasts; recursive forecasts; and direct forecasts. We consider both recursive and direct forecasts for generality as both have advantages and disadvantages, so it is not obvious a priori which will perform better (see Section 2.7.7 of Petropoulos et al (2022)). Model-based forecasts are re-estimated at each forecast step using expanding window estimation on the real-time data.

While our aim is to forecast the level of the exchange rates, we estimate the models using log levels. We take the natural log of the exchange rate, construct the no-change, autoregressive or direct forecast for the log of the period-average exchange rate, and then take the exponent of the forecast to convert back into the level of the period-average exchange rate. We do this because log variables are more likely to be closer to satisfying the assumptions of symmetric errors.^[6]

We denote daily, month-average and end-of-month exchange rates by D_t, A_m and Z_m respectively. We now denote log levels by lower case letters: d_t, a_m and z_m. We continue to assume that there are n days in each month to simplify our notation. We let M denote the current month, which means the forecaster has access to data from months m = 1,...,M when making a real-time forecast for a future month M+h.

4.2.1 No-change forecasts

We consider two types of no-change forecasts:

Month-average no-change. The forecast for the monthly average in any future month M+h is the last observed monthly average:

{\hat{a}}_{M + h | M} = a_{M} \forall h

End-of-month no-change. The forecast for the monthly average in any future month M+h is the current end-of-month level.

{\hat{a}}_{M + h | M} = z_{M} \forall h

This corresponds to the traditional assumption that the high-frequency series is a random walk. That is, if the daily exchange rate follows a random walk, then the expected period-average exchange rate in all future months equals the end-of-month no-change forecast (see Ellwanger and Snudden (2023); McCarthy and Snudden (forthcoming)). Note that the ‘daily no-change’ forecast, where our forecast for the month-average level in any future month M+h would be the latest daily level, d_Mn, is exactly equivalent to the end-of-month no-change forecast. This is because, in our forecast evaluation, the forecasts are always constructed at the end of each month.

4.2.2 Recursive AR(1) forecasts

We make recursive forecasts using autoregressive models of order 1 (AR(1)), estimated on exchange rate (log) levels using OLS.^[7] We consider the three ways to construct recursive forecasts of period-averages:

Recursive bottom-up. We estimate an AR(1) on daily exchange rates.

(5)

d_{t + 1} = α + β d_{t} + e_{t + 1} \forall t = 1, ..., M n - 1

We use this model to make recursive forecasts for the daily exchange rate for all future days. We then average those daily forecasts to obtain month-average forecasts (see Lütkepohl (1986); Benmoussa, Ellwanger and Snudden (forthcoming)).

{\hat{a}}_{M + h | M} = \frac{1}{n} \sum_{t = 1}^{n} {\hat{d}}_{(M + h - 1) n + t | M} \forall h

Recursive end-of-period. We estimate an AR(1) model of end-of-month exchange rates.

(6)

z_{m + 1} = α + β z_{m} + e_{m + 1} \forall m = 1, ..., M - 1

The recursive forecasts for the end-of-month exchange rates are then used as forecasts for the monthly average. The forecast of the end-of-period EER can be equal to the period average at short horizons when the underlying series is persistent and converges at longer horizons (Ellwanger et al 2023). Importantly, this allows us to quantify whether existing point forecasts in the literature will be good forecasts for forecasts of period-average exchange rates.

Recursive of month-average inputs. We estimate an AR(1) model of month-average exchange rates.

(7)

a_{m + 1} = α + β a_{m} + e_{m + 1} \forall m = 1, ..., M - 1

We then use this model to make recursive forecasts for all future horizons.

4.2.3 Direct forecasts

We construct direct forecasts using linear regressions estimated on exchange rate (log) levels. We consider the three ways to construct direct forecasts of period averages:

Direct UMIDAS. For each horizon h we estimate a regression of the month-average exchange rate in m+h on the latest daily observation known on the forecast date. We estimate the parameters of the model with OLS without any restrictions. This is the bottom-up direct forecast equivalent of Equation (5), see Lee and Snudden (2025), and unrestricted mixed data sampling (UMIDAS) (Foroni, Marcellino and Schumacher 2015). Since the latest daily observation available on the forecast date is the current end-of-month exchange rate, z_m, this model can be written:

(8)

a_{m + h} = α_{h} + β_{h} z_{m} + e_{m + h} \forall m = 1, ..., M - h

We then use the estimated model to directly forecast the month-average exchange rate.

{\hat{a}}_{M + h | M} = {\hat{α}}_{h} + {\hat{β}}_{h} z_{M}

Direct end-of-period. For each horizon, we estimate a regression of the end-of-month exchange rate in m+h on the end-of-month exchange rate in m.

(9)

z_{m + h} = α_{h} + β_{h} z_{m} + e_{m + h} \forall m = 1, ..., M - h

We then use this estimated model to produce an h-month-ahead forecast of the end-of-month exchange rate:

{\hat{z}}_{M + h | M} = {\hat{α}}_{h} + {\hat{β}}_{h} z_{M}

Again, the forecasts for the end-of-month exchange rates are then used as the forecasts for monthly average rates, ${\hat{a}}_{M + h} = {\hat{z}}_{M + h}$ .

Direct month-average inputs. For each horizon h, we estimate a regression of the month-average exchange rate in m+h on the month-average exchange rate in month m.

(10)

a_{m + h} = α_{h} + β_{h} a_{m} + e_{m + h} \forall m = 1, ..., M - h

We then use the estimated model to directly forecast the month-average exchange rate in h months.

{\hat{a}}_{M + h | M} = {\hat{α}}_{h} + {\hat{β}}_{h} a_{M}

Footnotes

For example, if we think it is equally likely that an exchange rate could appreciate by 1 per cent or depreciate by 1 per cent, then we should model the log exchange rate using a model with symmetric errors, rather than modelling the exchange rate itself as having symmetric errors. [6]

For a handful of countries, the estimated AR(1) model has a coefficient that is outside (–1,1), suggesting that exchange rates are non-stationary. Where this occurs, the country is excluded from our results. [7]