HPF is one of the most used methods and one of the most controversial. HPF minimizes the following objective function:

$\underset{{\left\{y_t^{*}\right\}}_{t=1}^T}{min}\{\sum _{t=1}^T{(y_t-y_t^{*})}^2+\lambda \sum _{t=1}^T{[(y_t^{*}-y_{t-1}^{*})-(y_{t-1}^{*}-y_{t-2}^{*})]}^2\},$ (1)

where: y_t is the observed time series (real GDP); y_t^* is the long-term unobserved component (HP trend or potential GDP), which is assumed to follow a smooth ­trajectory; λ is the smoothing parameter, which influences HPF results. When λ → 0, the HP trend, y_t^*, will be the original observed series, y_t. A considerable value for λ results in a smooth trend component, while in the extreme case of λ → ∞, the trend component will be a linear trend.

One of the major criticisms of HPF is its ad hoc choice for the value of the smoothing parameter. Following Hodrick and Prescott (1997), the smoothing parameter is set at 1,600 for quarterly data in this paper. Despite all the shortcomings, HPF remains the most straightforward method and is still primarily used in economics to estimate the potential GDP.

The idea behind BPF is that all stationary time series can be converted to frequency domain based on the spectral representation theorem. One can define a business cycle as fluctuations at certain frequencies. In the frequency domain, it is possible to analyze which cycles of different lengths contribute to the dynamics­ of the time series. Typically, researchers define business cycles between six quarters and thirty-two quarters of frequency. Therefore, business cycles can be estimated by eliminating cycles outside this range.

The version of the BPF developed by Baxter and King (1995) is calculated by moving averages. Thus, missing observations are generated at the beginning and the end of the sample period. We use the Christiano and Fitzgerald (2003) version to avoid the loss of observations. This is an asymmetric filter where the weights on the leads and lags can differ. The asymmetric filter is time-varying, with the weights depending on the data and changing for each observation. Thus, it avoids observation losses at the sample period’s beginning and end.

MVHPF is developed by Laxton and Tetlow (1992). This method adds equations derived from known economic relationships to HPF. Chagny and Döpke (2001) provide the following example:

π_t = π_t^e + A (L)(y_t – y_t^*) + ε_π,t. (2)

Equation (2) is an augmented Philips-curve relationship, where the actual inflation rate π depends on inflation expectations (π^e) and the current and lagged output gap. The squared residuals from the Philips-curve expression are added to the objective function of HPF, leading to the following minimization problem:

$min\sum _{t=1}^T{(y_t-y_t^{*})}^2+\lambda \sum _{t=2}^T{[(y_t^{*}-y_{t-1}^{*})-(y_{t-1}^{*}-y_{t-2}^{*})]}^2+\sum _{t=1}^T{ß}_t{\epsilon }_{\pi ,t}^2,$ (3)

where, when we assume that A (L) = α₂ + α₃ L, so that only the contemporaneous and the one-period lagged values of the output gap is included in equation (2):

$\sum _{t=1}^T{ß}_t{\epsilon }_{\pi ,t}^2=\sum _{t=1}^T{ß}_t{[{\pi }_t-({\vartheta }_1+a_1{\pi }^e+a_2(y_t-y_t^{*})+a_3(y_{t-1}-y_{t-1}^{*}))]}^2.$ (4)

An important objective of the multivariate filter is to reduce the uncertainty associated with estimates of the potential output (y_t^*). Thus, the smaller the variance­ of the residuals, the more valuable the information added by the economic relationship will be. Following Laxton and Tetlow (1992), we set β_t = 1 and λ = 1,600. The initial values for the parameters ϑ₁, α₁, α₂, and α₃ are estimated by a multivariate regression for inflation.

A classic example of a SVAR is presented in the seminal paper of Blanchard and Quah (1989). Building from Blanchard and Quah (1989) and following Funke (1997), we applied a bivariate VAR model including real GDP growth rate and inflation rate:

$y_t=\left[\begin{array}{l}\mathrm{∆}\log \left(q_t\right)\\ \mathrm{∆}\log \left(p_t\right)\end{array}\right]~I(0),$ (5)

where y_t is a two-variable vector with real GDP growth rate (∆log(q_t)) and inflation rate (∆log(p_t)).

Both variables are assumed to be stationary. The moving average representation of the underlying SVAR model can be written as follows:

$\left[\begin{array}{l}\mathrm{∆}\log \left(q_t\right)\\ \mathrm{∆}\log \left(p_t\right)\end{array}\right]=\sum _{i=0}^{\mathrm{\infty }}{L}^i{A}_i{\epsilon }_t$ (6)

where: A_i is a matrix; L is the lag operator; ε_t is white noise residuals capturing supply and demand shocks.

To identify the type of structural shocks driving the system, a long-run neutrali­ty restriction is imposed, whereby the matrix of long-run multipliers A (1) is forced to be upper triangular:

$\sum _{i=0}^{\mathrm{\infty }}{A}_{11,i}=0.$ (7)

This is the case when the cumulative amount of (temporary) effects of demand shocks on changes in income is zero, and thus demand shocks do not have long‑run effects on the income level. The output gap within this framework is given by the fraction of GDP movements explained by demand shocks. The potential GDP is given by the deterministic component of the model and the impact of supply shocks.

The recursive estimation method is used to construct the out-of-sample forecasts. For the out-of-sample forecast, the model parameters are estimated only up to the start date of the forecast. We consider the period 2000Q1 to 2022Q4, in which the countries had more stable prices following the pre‑2000 introduction of inflation targeting (Brazil, India, and South Africa) and a tolerance­ band for China and Russia. Hence, the first date of the estimation sample is 2000Q1. We make forecasts up to 12 quarters ahead and evaluate these forecasts in the 2010Q1–2022Q4 period. According to the recursive estimation method, the first observation of the estimation period is unchanged, but we keep lengthen­ing the estimation sample period. We first estimate the model for 2000Q1–2009Q4 and forecast for 2010Q1–2012Q4. The second estimation period is the 2000Q1–2010Q1 period, and forecasts for 2010Q2–2013Q1 are made, and so on. We re-estimate the coefficients {α, α_1j, α_2j, and ρ_1j} at each forecast round by ordinary least squares. We compare our forecasts with the realized inflation values to calculate forecast errors.

We have four gap models for inflation forecasting (equation 8), corresponding to the number of alternative output gap estimates used in this paper. Since we forecast multi-periods-ahead using a model that includes the output gap, we also must forecast the output gap. For example, for the out-of-sample inflation forecast we make from 2012Q4 to 2013Q1 using data up to 2012Q4, we need to know the 2013Q1 forecast value of the output gap. The following univariate autoregressive model is used to forecast the output gap:

$ga{p}_t=a+\sum _{j=1}^m{\rho }_{1j}ga{p}_{t-j}+u_t.$ (9)

We first calculate one-period ahead forecasts from equations (8) and (9) and then iterate these forecasts using the two equations over the forecast horizon (h) to obtain multi-period ahead forecasts (this is the so-called iterated forecast method, see, for example, Pincheira and West, 2016).

To compare the forecasting performance of the output gap models with a benchmark model, we use a simple autoregressive forecasting model of inflation, henceforth “Inflation AR-benchmark model”.

${\pi }_t=a+\sum _{i=1}^a{\omega }_{1i}{\pi }_{t-i}+u_t$ (10)

We employ well-known loss functions to evaluate the forecasts: mean error (ME), mean absolute error (MAE), root mean squared error (RMSE), and mean squared forecast error (MSFE), for which we calculate the forecast error in percentages. That is, since inflation is defined as the difference in the logarithm of the price level, π_t = ∆log(p_t), we calculate price level forecasts and express the forecast error as a percent of the price level:

${\epsilon }_{t+h|t}=\frac{p_{t+h}-p_{t+h|t}}{p_{t+h}},$ (11)

where: ε_t+h|t is the h-period ahead forecast error expressed as the percent of the h‑period head price level; p_t+h|t is the h-period ahead forecast of the price level using information up to time t; p_t+h is the actual price level at time t + h.

Our out‑of‑sample evaluation period, 2010Q1–2022Q4, is sufficiently long and includes critical economic developments for emerging markets, such as the recovery from the global financial crisis, the impact of the 2013 taper tantrum, and the sharp decline in economic growth due to COVID-19 pandemic.

We compare the forecast accuracy of the output gap model to the Inflation AR-benchmark model. This is done by testing the null hypothesis that the difference between the output gap model’s MSFEs and that of the AR model is zero. The alternative hypothesis is that the output gap models forecast better than the Inflation AR-benchmark model. The Clark and West (2007) approach is used for the test.

We also test for the unbiasedness of the forecasts. We estimated the following equation for the forecast unbiasedness test, Clements et al. (2007):

log(p_t+h) – log(p_t+h) = β + ε_t, (12)

where β is a parameter to estimate; ε_t is the regression error. The null hypothesis of unbiasedness corresponds to the test of β = 0. We regress forecast error on a constant using ordinary least squares.

Data covers the period from 2000Q1 to 2022Q4. The data collected for each country includes the following variables: real GDP, inflation, and inflation expectations. The data is sourced from the central banks of the respective countries, the World Bank Database, the OECD, IMF International Financial Statistics, and BIS databases. The quarterly GDP data is seasonally adjusted. The average growth rate declined drastically post the global financial crisis, except for India. The inflation rate has remained relatively stable in four countries, except Russia, where it fell to less than half post-GFC compared to pre-GFC (see Table 1). The variability over the sample period suggests there could have been changes in the size of shocks to trend and cyclical components of these economies, particularly in South Africa.

For all five countries, output gap estimates from the different methods show remarkably similar dynamics over time, with a sharp decline during the onset of the COVID-19 pandemic in 2020 (see Fig. 1). In addition, we observe a V-shaped recovery post-COVID-19 pandemic, but the growth rates remain below pre-pandemic levels.

Figure 1.

Output gap estimates.

Note: BCB — Bank of Brazil; SARB — South African Reserve Bank; RHS — right-hand scale. The attempts to obtain central bank output gap estimates for India, China and Russia were unsuccessful. Source: Author’s calculations.

The methods were also able to capture the significant recession periods. For Brazil, 2001, 2003, and 2008 show negative output gaps. The energy crisis can explain the 2001 recession, the high interest rates, and the substantial external economic slowdown (Considera et al., 2019). The recession from 2003 to early 2004 can be explained by the low-risk appetite of foreign investors following the election of the new president at the end of 2002. The other year with negative output gaps is 2008 due to a well-known shock of the global financial crisis.

For China, all methods captured large negatives in the output gap in 2003Q1, which can be explained by the economic impact of the Severe Acute Respiratory Syndrome (SARS) outbreak.

In the case of India, the notable negative output gap from 2002 to 2003 can be explained by a large drop in agricultural production following the worst drought to hit the country at the time (Nagaraj, 2013). The economy rebounded strongly until the global financial crisis and growth was well above its potential before the pandemic in 2020.

The models, except the SVAR, suggest that the Russian economy was overheating before the global financial crisis (GFC) in 2008. The SVAR model, however, shows a divergence from the other models during the GFC. Notably, the SVAR model captures the underperformance of the economy already in the 2008Q1, whereas other models start capturing economic slowdown in 2008Q3. The SVAR model is designed to capture structural shocks in the econo­my such as external shocks and the structural interactions between multiple economic variables. In the absence of a unique structural shift in the Russian economy in 2008, the SVAR model’s capture of the 2008 recession early on can be attributed to the model’s sensitivity to structural shocks, and the ability to model dynamic interactions among multiple variables. We observe similar results during the COVID-19 in 2020.

The univariate models suggest that post the 2008 GFC, the economy performed well around its potential. In stark contrast, the MVHPF model suggests overwhelming growth, to levels above those seen pre-GFC. This discrepancy could indicate that either the MVHPF is overestimating the output gap or the HPF and BPF are underestimating it.

The underperformance of the South African economy is not surprising as the country’s economy has been constrained by multiple factors such as skilled labor shortages, structurally low domestic savings and investment, lack of fiscal discipline, previous unproductive investments, and rising import of consumption goods (Binatli and Sorjahbji, 2012, Matthee, 2016, Purifield et al., 2014). Fedderke and Mengisteab (2016) find similar results.

The output gap estimates reported by the central banks of Brazil and South Africa are similar to our results. The alignment of our estimates with those of central banks implies that our simpler models can approximate the result from the more complex models used by central banks. Policymakers and researchers can use our results with greater confidence, knowing they are consistent with the central banks’ assessments.

Therefore, contrary to the widespread view that using various models is crucial for estimating the output gap, our results suggest that the choice of measure might not be of particular importance (see Fig. 1). Generally, the high level of similarity of alternative output gap estimates between the models is a common finding in the existing literature (Chagny and Döpke, 2001; Altar et al., 2010; Saulo et al., 2010; Kemp, 2014, Fedderke and Mengisteab, 2016; Pham, 2020).

Tables 2 to 6 display the correlation of our output gap estimates from the different­ models to analyze the similarities or lack thereof among them. A strong and positive correlation between alternative estimates suggests that the models­ capture the deviation of actual output above or below potential output in a similar manner for the countries and time periods analysed in this paper. This indicates strong similarities in the way the models interpret data. Conversely, low or negative correlations suggest a lack of consensus and significant differences in output gap estimates.

The correlation coefficients between HPF and BPF results are relatively high, between 0.67–0.96. However, the MVHPF and SVAR estimates have low correlation with HPF and BPF estimates, probably reflecting the different methodological assumptions. The correlations are positive and strong between our methods (especially the HPF) and the central bank estimates for Brazil and South Africa. This indicates similarities between the methods we used and those employed by the central banks of Brazil and South Africa.

The low correlations imply different characteristics of the business cycle. Chagny and Döpke (2001) find similar results for the eurozone and report that although the estimates exhibit similar dynamics, the statistical comparison (like correlations) suggests stark contrasts.

The null hypothesis is that the output gap has no information content about inflation (see α_2j = 0 in equation 8). If the null hypothesis cannot be rejected, then the output gap does not influence current and future inflation and hence does not help inflation forecasts. In Tables 7 to 11, the statistical significance of gap models suggests that the output gap is useful for inflation forecasting. For four of the five BRICS countries except Russia, at least one of the gap models is helpful for inflation forecast. In fact, for South Africa, all the output gap models perform well. In addition, the R-squared values show that the output gap measures have moderate explanatory power for the inflation of South Africa. These findings are consistent with some of the existing studies focusing on the South African economy (Akinboade, 2005; Fedderke and Mengisteab, 2016). Similar findings for India are reported by Virmani (2004).

The estimated coefficient on lagged inflation has the expected sign, except for China. For instance, in the case of South Africa, a one percentage point increase in price levels in a given quarter is predicted to increase the price levels in the next quarter by 0.51 when the BPF is used, while the same coefficient is 0.22 for HPF, 0.26 for MVHPF and 0.002 for SVAR (see Table 11). However, we note that the esti­mated coefficient for lagged inflation is not statistically significant for China, India, and Russia. The coefficients of the output gaps also have the expected sign. A one percentage point increase in the output gap is predicted to induce an increase in inflation in the short run. The estimated sign is negative only in the case of MVHPF for Brazil and Russia, although these estimates are statistically not significant.

Our results for Russia are contrary to the only other study we know, Kloudova (2015), which assessed the output gap for Russia and found the HPF method to perform well for the Russian economy. Although the HPF has the expected positive coefficient sign, it is insignificant. In the case of China, Gerlach and Peng (2006) shows that the standard Phillips curves tend not to fit the country’s data well likely due to omitted structural and institutional variables, such as price deregulation, trade liberalization, and changes in the exchange rate regime.

Tables 12 to 14 present the percentage point averages for mean error (ME), mean absolute error (MAE), and root mean squared error (RMSE) obtained in an iterative procedure described in Section 3. We consider the 2010Q1–2022Q4 out-of-sample period. A negative value of the forecast error means the actual value for inflation is lower than its forecast value. Analogously, the positive value of the forecast error means the actual value is higher than the forecast. The ME indicator is useful to check whether the average of the forecast errors is close to zero, while MAE and RMSE indicators treat positive and negative forecast errors similarly and thus do not depend on the sign of the error.

According to MAE and RMSE, the inflation AR-benchmark model produces smaller forecast errors than the gap models in most cases (Tables 13 and 14). However, compared to the inflation AR-benchmark model, the MVHPF and HPF models appear to be good models for forecasting inflation for India and South Africa, respectively. Moreover, amongst the gap models, the MVHPF produces smaller forecast errors in most cases. The ME shows a general consistency in terms of the direction (the sign) of the errors between the different models. There are very few cases where the inflation AR-benchmark model and gap models are producing conflicting signs.

Table 15 shows the ratio of MSFEs from the three gap models and the AR-benchmark model for each country, expressed as percent of the AR‑benchmark model. Thus, an MSFE value below one hundred indicates that the output gap model outperforms the AR-benchmark model over the period 2010Q1–2022Q4. Clark and West (2007) test the null hypothesis that the MSFEs of a larger model and a nested benchmark model are equal. We use their test to calculate the p‑values of the null hypothesis of equal forecast accuracy against the one-sided alternative that the output gap model is better than the benchmark model. We report these p-values in parenthesis in Table 15.

The Clark and West (2007) test shows diverse results across BRICS economies. For Brazil, the AR-benchmark model performs better than the gap models in the longer forecast horizon (twelve quarters ahead, 12Q) than in the short-term (one-quarter and four-quarters ahead, 1Q and 4Q). The p-values are consistent with MSFEs, as we reject the null hypothesis which states that the gap and AR‑benchmark model are equal. Moreover, the MVHPF model is less accurate than the HPF and BPF models for the Brazilian economy. We find a similar case for the Russian economy, and the gap models are not performing well in all forecast horizons. This result is consistent with the findings from the Phillips-curve estimation.

For China, India and South Africa, the gap models are generally more useful than the AR-benchmark model in forecasting inflation in most forecast horizons. However, there are specific variations in the results. For China, the AR benchmark performs better in short forecast horizons (1Q and 4Q), but it is still surpassed by the HPF model, which performs well even compared to other gap models in all forecast horizons. In the case of South Africa, the HPF is the worst-performing model in longer forecast horizons (12Q). For India, the MVHPF is performing well in 1Q and 12Q forecast horizons under our evaluation.

We have found that there is no one size fits all model. Based on our overall­ results we cannot claim that gap models perform better than the Inflation AR‑benchmark model. Nor we can claim outright that the benchmark model is a better measure. We may assert that to an extent our results reflect the diverse economic structures of the BRICS economies.

Similarly to existing research, the inflation equation (the Philips curve) suggests that both lagged inflation and the output gap help forecast inflation. The ­importance of the role of backward-looking information in the inflation ­dynamics is underscored by studies that have used a hybrid New Keynesian Phillips curve (NKPC) to capture the true data-generating process of inflation (Hubert and Mirza, 2019). The NKPC aims to establish the role of backward- and forward-looking information in the inflation expectation formation process. The general finding from the literature, however, is that the influence of backward-looking information tends to diminish over time (Hubert and Mirza, 2019).

The MSFE evaluation method shows that the statistical significance of the ­results varies with the forecast horizon for the studied sample periods of the BRICS countries. For Brazil, the Inflation AR-benchmark model is more reliable for a longer forecast horizon. For China, the benchmark model is more reliable in short forecast horizons. For Russia, the benchmark model is the best-performing model for all forecast horizons. For India and South Africa, the gap models are more accurate than the benchmark model in almost all forecast horizons.

Generally, there were notable fluctuations in inflation in BRICS countries over the period 2000–2022. Nonetheless, the gap models have performed much better for countries that have implemented inflation targeting, India and South Africa in particular, which had the second and third lowest inflation variability (after China) among the five countries. This result might suggest that the gap models are more accurate in forecasting inflation under the environment of increased inflation stability and monetary policy transparency.

When we compare the gap models to each other, again it depends on which economy you are looking at. The MSFE gives the impression that the MVHPF and SVAR model are better in most cases. However, this result does not hold for all the different forecast horizons in the respective countries. For example, MVHPF is the best-performing model in short forecast horizons for Brazil and Russia and performs better in the longer term for India.

Increased stability in inflation which is possible due to inflation targeting in Brazil, and India could explain why MVHPF performs better than BPF and HPF for these countries. Recall that the MVHPF model can incorporate inflation expectations, for which the literature has shown that expectations tend to be well anchored under the inflation targeting framework (Suh and Kim, 2021). In addition, the MVHPF model possibly captures the business cycle movements much better than HPF and BPF, as the model incorporates current and lagged output gap changes in estimating inflation. Since the BRICS countries have experienced varying business cycles, the differences in terms of the role of gap models in forecasting inflation should not be surprising (see discussion in Section 2). This discussion reinforces the argument that univariate models may omit helpful information contained in economic relationships.

Table C1 in Appendix C reports p-values for the hypothesis test, in which the null hypothesis states that the forecasts are unbiased. We start from a 1-quarter ahead forecast, then a 2-quarter ahead forecast, a 3-quarter ahead forecast, and so on. For each forecast horizon, we test the unbiasedness of the forecasts given by each of the four methods (BPF, HPF, MVHPF and SVAR). For HPF MVHPF and SVAR methods, the null hypothesis of unbiasedness cannot be rejected (­considering p‑values­ larger than 10%) in most cases, particularly between 1-quarter to 10-quarters ahead. The BPF method rejects the null hypothesis at a 1% level of significance, except for Brazil and China. In other words, we find the BPF forecasts to be biased. In addition, the SVAR becomes biased from the 11- to 20-quarters ahead.

We check the robustness of the results for inflation forecasting to alternative sample periods (such as the post-global financial crisis period after 2010Q1) and the method to calculate out-of-sample forecasts (rolling forecast scheme with various window sizes, ranging from five years to thirty-one years, instead of our default recursive scheme). The results are not considerably different from our initial analysis.