## Tuesday, February 25, 2020

### Beveridge-Nelson Filter

Authors and guest post by Benjamin Wong (Monash University) and Davaajargal Luvsannyam (The Bank of Mongolia)

Analysis of macroeconomic time series often involves decomposing a series into a trend and cycle components. In this blog post, we describe the Kamber, Morley, and Wong (2018) Beveridge-Nelson (BN) filter and the associated EViews add-in.

### Introduction

In this blog entry, we will discuss the Beveridge-Nelson (BN) filter - the Kamber, Morley, and Wong (2018) modification of the well-known Beveridge and Nelson (1981) decomposition. In particular, we will discuss the application of both procedures to estimating the output gap, which the US Bureau of Economic Analysis (BEA) and the Congressional Budget Office (CBO) define as the proportional deviation of the real actual gross domestic product (GDP) from the real potential GDP.

The analysis to follow will use quarterly data from the post World War II period 1947Q1 to 2019Q3 and will be downloaded from the FRED database. In this regard, we begin by creating a new quarterly workfile as follows:
1. From the main EViews window, click on File/New/Workfile....
2. Under Frequency select Quarterly.
3. Set the Start date to 1947Q1 and the set the End date to 2019Q3
4. Hit OK.
Next, we fetch the GDP data as follows:
1. From the main EViews window, click on File/Open/Database....
2. From the Database/File Type dropdown, select FRED Database.
3. Hit OK.
4. From the FRED database window, click on the Browse button.
5. Next, click on All Series Search and in the Search for box,type GDPC1. (This is the real actual seasonally adjusted GDP)
6. Drag the series over to the workfile to make it available for analysis.
7. Again, in the Search for box, type GDPPOT. (This is the real potential seasonally unadjusted GDP estimated by the CBO)
8. Drag the series over to the workfile to make it available for analysis.
9. Close the FRED windows as they are no longer needed.
 Figure 1a: FRED Browse Figure 1b: FRED Search

Next, rename the series GDPC1 to GDP by issuing the following command:
    rename gdpc1 gdp

We now show how to obtain the implied estimate of the output gap from the CBO to provide the user some perspective on how to obtain the output gap. In particular, the CBO implied estimate of the output gap is defined using the formula: $$CBOGAP = 100\left(\frac{GDP - GDPPOT}{GDPPOT}\right)$$ For reference, we will create this series in EViews and call it CBOGAP. This is done by issuing the following command:
    series cbogap = 100*(gdp-gdppot)/gdppot

We also plot CBOGAP below:
 Figure 2: CBO implied estimate of the output gap

### BN Decomposition

Recall here that for any time series $y_{t}$, the BN decomposition determines a trend process $\tau_{t}$ and a cycle process $c_{t}$, such that $y_{t} = \tau_{t} + c_{t}$. In this regard, the trend component $\tau_{t}$ is the deviation of the long-horizon conditional forecast of $y_{t}$ from its deterministic drift $\mu$. In other words: $$\tau_{t} = \lim_{h\rightarrow \infty} E_{t}\left(y_{t+h} - h\mu\right) \quad \text{where} \quad \mu = E(\Delta y_{t})$$ On the other hand, the cyclical component is the deviation of the underlying process from its long-horizon forecast. Intuitively, when $y_{t}$ represents the GDP of some economy, the cycle process $c_{t} = y_{t} - \tau_{t}$ is interpreted as the output gap.

In practice, in order to capture the autocovariance structure of $\Delta y_{t}$, the BN decomposition starts by first fitting an autoregressive moving-average (ARMA) model to $\Delta(y)$ and then proceeds to derive $\tau_{t}$ and $c_{t}$. For instance, when the model of choice is AR(1), the BN decomposition derives from the following steps:

1. Fit an AR(1) model to $\Delta y_{t}$: $$\Delta y_{t} = \widehat{\alpha} + \widehat{\phi}\Delta y_{t-1} + \widehat{\epsilon}_{t}$$
2. Estimate the deterministic drift as the unconditional mean process: $$\widehat{\mu} = \frac{\widehat{\alpha}}{1 - \widehat{\phi}}$$
3. Estimate the BN trend process: $$\widehat{\tau}_{t} = \left(y_{t} + \left(\frac{\widehat{\phi}}{1 - \widehat{\phi}}\right) \Delta y_{t}\right) - \left(\frac{\widehat{\phi}}{1 - \widehat{\phi}}\right) \widehat{\mu}$$
4. Estimate the BN cycle component: $$\widehat{c}_{t} = y_{t} - \widehat{\tau}_{t}$$

As an illustrative example, consider the BN decomposition of US quarterly real GDP. To conform with the Kamber, Morley, and Wong (2018) paper, we will also transform the raw US real GDP as 100 times its logarithm. In this regard, we generate a new EViews series object LOGGDP by issuing the following command:
    series loggdp = 100 * log(gdp)

At last, following the 4 steps outlined earlier, we derive the BN decomposition in EViews as follows:
    series dy = d(loggdp)
equation ar1.ls dy c dy(-1) 'Step 1
scalar mu = c(1)/(1-c(2)) 'Step 2
series bntrend = loggdp + (dy - mu)*c(2)/(1 - c(2)) 'Step 3
series bncycle = loggdp - bntrend 'Step 4

The BN trend and cycle series are displayed in Figure 2 below.

 Figure 3a: BN Trend Figure 3b: BN Cycle

To see how the BN decomposition estimate of the output gap compares to the CBO implied estimate of the output gap, we plot both series on the same graph.

 Figure 4: BN Cycle vs CBO implied output gap estimate

Evidently, the BN cycle series lacks persistence (very noisy), lacks amplitude (low variance), and in general, does not exhibit the characteristics found in the CBO implied estimate of the output gap, CBOGAP.

### The BN Filter

First, to explain why the BN estimate of output gap lacks the persistence of its true counterpart, recall the formula for the BN cycle component for an AR(1) model: $$c_{t} = y_{t} - \tau_{t} = -\frac{\phi}{1-\phi}(\Delta y_{t} - \mu)$$ Clearly, when $\phi$ is small, $\Delta y_{t}$ is not very persistent. Since $c_{t}$ is only as persistent as $\Delta y_{t}$, the cycle component itself lacks the persistence one expects of the true output gap series.

Next, to explain why $c_{t}$ lacks the expected amplitude, define the signal-to-noise ratio $\delta$ for any time series as the ratio of the variance of trend shocks relative to the overall forecast error variance. In other words: $$\delta \equiv \frac{\sigma^{2}_{\Delta \tau}}{\sigma^{2}_{\epsilon}} = \psi(1)^{2}$$ which follows since $\Delta\tau = \psi(1)\epsilon_{t}$ and $\psi(1) = \lim_{h\rightarrow \infty} \frac{\partial y_{t+h}}{\partial \epsilon_{t}}$. Intuitively, $\psi(1)$ is the long-run multiplier that captures the permanent effect of the forecast error on the long-horizon conditional expectation of $y_{t}$. Quite generally, as demonstrated in Kamber, Morley, and Wong (2018), for any AR(p) model: \begin{align} \Delta y_{t} = c + \sum_{k=1}^{p}\phi_{k}\Delta y_{t-k} + \epsilon_{t} \label{eq1} \end{align} the signal-to-noise ratio is given by the relation \begin{align} \delta = \frac{1}{(1-\phi(1))^{2}} \quad \text{where} \quad \phi(1) = \phi_{1} + \ldots + \phi_{p}\label{eq2} \end{align} In particular, when the forecasting model is AR(1), as was the case in the BN decomposition above, the signal-to-noise ratio is simply $\delta = \frac{1}{(1-\phi)^{2}}$ and in the case of the US GDP growth process, it is $\delta = \frac{1}{(1-0.36)^{2}} = 2.44$. In other words, the BN trend shocks exhibit higher volatility than quarter-to-quarter forecast errors and the signal-to-noise ratio is therefore relatively high. In fact, in the case of a freely estimated AR$(p)$ model of output growth, $\phi(1) < 1$, which implies that $\delta > 1$. In other words,the trend will be more volatile than the cycle, and at odds if one expects the cycle shocks (the output gap amplitude) to explain the majority of the systematic forecast variance.

To correct for the aforementioned shortcomings of the BN decomposition, Kamber, Morley, and Wong (2018) exploit the relationship between the signal-to-noise ratio and the AR coefficients in equation \eqref{eq2}. In particular, they note that equation \eqref{eq2} implies that: \begin{align} \phi(1) = 1 - \frac{1}{\sqrt{\delta}} \end{align} In this regard, the idea underlying the BN filter is to fix a specific value to the signal-to-noise ratio, say $\delta = \bar{\delta}$. Subsequently, the BN decomposition is derived from an AR model, the AR coefficients of which are forced to sum to $\bar{\phi}(1) \equiv 1 - \frac{1}{\sqrt{\bar{\delta}}}$. In other words, the BN decomposition is derived while imposing a particular signal-to-noise ratio.

It is important to note here that estimation of the BN decomposition under a particular signal-to-noise ratio restriction is in fact straightforward and does not require complicated non-linear routines. To see this, observe that equation \eqref{eq1} can be rewritten as: \begin{align} \Delta y_{t} = c + \rho \Delta y_{t-1} + \sum_{k=1}^{p-1}\phi^{\star}_{k}\Delta^{2} y_{t-k} + \epsilon_{t} \label{eq3} \end{align} where $\rho = \phi_{1} + \ldots + \phi_{p}$ and $\phi^{\star}_{k} = -\left(\phi_{k+1} + \ldots + \phi_{p}\right)$. Then, imposing the restriction $\rho = \bar{\rho} \equiv \bar{\phi}(1)$ reduces the regresion in \eqref{eq3} to: \begin{align} \Delta y_{t} - \bar{\rho} \Delta y_{t-1} = c + \sum_{k=1}^{p-1}\phi^{\star}_{k}\Delta^{2} y_{t-k} + \epsilon_{t} \label{eq4} \end{align} In other words, $\bar{\rho}\Delta y_{t-1}$ is brought to the left hand side and the regressand in the regression \eqref{eq4} becomes $\Delta \bar{y}_{t} \equiv \Delta y_{t} - \bar{\rho} \Delta y_{t-1}$.

### Why Use the BN Filter?

Before we demonstrate the BN Filter add-in, we quickly outline two reasons why the BN filter might be a reasonable approach, particularly when estimating the output gap.
1. When analyzing GDP growth, standard ARMA model selection often favours low order AR variants, which, as discussed earlier, produce high signal-to-noise ratios.
2. Unlike alternative low signal-to-noise ratio procedures such as deterministic quadratic detrending, the Hodrick-Prescott (HP) filter, and the bandpass (BP) filter, which often require large number of estimation revisions (as new data comes in) and are typically unreliable in out-of-sample forecasts (see Orphanides and Van Norden, (2003)), Kamber, Morley and Wong (2018) argue that the BN filter exhibits better out-of-sample performance and generally requires fewer estimation revisions to match observable data characteristics.

To further drive this latter point, we demonstrate the impact of ex-post estimation of the output gap using the HP filter. In particular, we will first estimate the output gap (the cycle component) of the LOGGDP series for the period 1947Q1 to 2008Q3 and call it HPCYCLE, and then again for the period 1947Q1 to 2019Q3 and call it HPCYLCE_EXPOST.

To estimate the HP filter cycle component for the period 1947Q1 to 2008Q3, we first set the sample accordingly by issuing the command:
    smpl @first 2008Q3

Next, we estimate the HP filter cycle series as follows:
1. From the workfile, double click on the series LOGGDP to open the series.
2. In the series window, click on Proc/Hodrick-Prescott Filter...
3. In the Cylce series text box, type hpcycle.
4. Hit OK.
 Figure 5: HP Filter

The steps are repeated for the sample period 1947Q1 to 2019Q3. A plot of both cycle series on the same graph is presented below.

 Figure 6: HP Cycle vs HP Cycle Ex Post

Evidently, the ex-post HP filter estimation of the output gap diverges from its shorter period counterpart starting from 2006Q1. It is precisely this drawback that we will see is not nearly as pronounced in BN filter estimates.

### BN Filter Implementation

To implement the BN Filter, we need to download and install the add-in from the EViews website. The latter can be found at https://www.eviews.com/Addins/BNFilter.aipz. We can also do this from inside EViews itself:
2. Click on the the BNFilter add-in.
3. Click on Install.

At last, we will demonstrate how to apply the BN Filter add-in using an AR(12) model. To do so, proceed as follows:
1. From the workfile window, double click on LOGGDP to open the spreadsheet view of the series.
2. To access the BN filter dialog, click on Proc/Add-ins/BN Filter
3. Stick with the defaults and hit OK.

 Figure 6: BN Filter Dialog

The signal-to-noise ratio, while not specified above, is chosen using the Kamber, Morley, and Wong (2018) automatic selection procedure which balances the trade off between fit and amplitude. Typically, the signal-to-noise ratio for the US using such a procedure is about 0.25, which implies a quarter of the shocks to US GDP are permanent. Below, we show the BN Filter cycle series both alone and in comparison to the CBO implied estimate of the output gap CBOGAP.

 Figure 7a: BN Filter Cycle Figure 7b: BN Filter Cycle vs CBO implied output gap estimate

We also plot a comparison of the BN Filter cycle series with the HP filtered cycle.

 Figure 8: BN Filter Cycle vs HP Filter Cycle

As we can see, the BN filter estimate of the the US output gap using an AR(12) model resembles what we would get for an output gap that has a low signal-to-noise ratio. The amplitude is reasonably large, we see business cycles, and the troughs line up with the recessions dated by the NBER. The amplitude of the output gap estimated using the BN Filter is comparable to that of the cycle obtained by the HP filter, as well as the implied estimated of the CBO, which is unlike what we see in Figure 4.

The BN filter add-in also accommodates the ability to incorporate knowledge about structural breaks. In particular, we will use 2006Q1 as a structural break which is consistent with the date found by a Bai and Perron (2003) test, used by Kamber, Morley and Wong (2018), and is consistent with independent work by Eo and Morley (2019). The following steps demonstrate the outcome:
1. From the workfile window, double click on LOGGDP to open the spreadsheet view of the series.
2. To access the BN filter dialog, click on Proc/Add-ins/BN Filter
3. Select the Structural Break box.
4. In the Date of structural break text box, enter 2006Q1.
5. Hit OK.

 Figure 9: BN Filter Cycle (Structural Break)

Now we see a more positive output gap post-2006 as the structural break accounts for the fact that the average GDP growth rate has fallen.

Suppose however that we were ignorant about the actual date of the break. This might be the case in practice as it could take a decade or more before one could empirically identify a structural break date. In this case, a possible option is to use a rolling window for the average growth rate. In this example, we use a backward window of 40 quarters as the average growth rate. The idea is that if there were breaks, they would be reflected in this window. When this is the case, we proceed as follows:
1. From the workfile window, double click on LOGGDP to open the spreadsheet view of the series.
2. To access the BN filter dialog, click on Proc/Add-ins/BN Filter
3. Select the Dynamic mean adjustment box.
4. Hit OK.

 Figure 10a: BN Filter Cycle (Dynamic Mean Adjustment) Figure 10b: BN Filter Cycle (Known vs Unknown Structural Break)

Evidently, the estimated output gap looks similar to the one estimated with an explicit structural break in 2006Q1. In general, this suggests that using a backward window to adjust for the mean growth rate might be a useful real-time strategy for dealing with breaks.

Users are not constrained to the automatic option, which balances the trade off between fit and amplitude. The BN filter add-in also allows users to specify a desired signal-to-noise ratio. For instance, the following example compares the difference in setting the signal-to-noise ratio $\delta$, to 0.05 (which implies 5% of the variance is permanent), against the default 0.25 which we derived earlier by leaving $\delta$ unspecified, and so uses the procedure which balances the trade off between fit and amplitude. To do so, we proceed as follows:
1. From the workfile window, double click on LOGGDP to open the spreadsheet view of the series.
2. To access the BN filter dialog, click on Proc/Add-ins/BN Filter
3. Select the Structural Break box.
4. In the Date of structural break text box, enter 2006Q1.
5. Hit OK.

The plot below summarizes the exercise.

 Figure 11: BN Filter Cycle (delta = 0.25 vs delta = 0.05

Unsurprisingly, specifying $\delta = 0.05$ results in an output gap with a larger amplitude than the default as the new specification implies a smaller proportion of the shocks to the forecast error is parsed to the trend, and so a larger proportion is parsed to the cycle, leading to a larger amplitude cycle.

Finally, we come back to the issue of revision. As we mentioned earlier, the BN filter should produce output gaps that are less revised as long as the AR forecasting model is stable, especially when compared to the heavily revised HP Filter. Here, we show the output gap estimated using the BN filter with data up to 2008Q3, and one ex-post up to 2019Q3. Clearly, the output gap is hardly revised, which address a key critique of Orphanides and Van Norden (2003).

 Figure 11: BN Filter Cycle (Ex-Post)

### Conclusion

In this blog post we have outlined the BN filter add-in associated with the work of Kamber, Morley and Wong (2018). In general, we hope the ease of using the add-in, together with some of the useful properties of the BN Filter will encourage practitioners to explore using the procedure in their work.

### References

1. Bai J. and Perron P.: Computation and analysis of multiple structural change models Journal of Applied Econometrics, 18(1) 1–22, 2003.
2. Beveridge S. and Nelson C. R.: A new approach to decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle Journal of Monetary Economics, 7(2) 151–174, 1981.
3. Eo Y. and Morley J.: Why has the US economy stagnated since the Great Recession University of Sydeny Working Papers 2017-14, 2019.
4. Kamber G., Morley J., and Wong B.: Intuitive and reliable estimates of the output gap from a Beveridge-Nelson filter The Review of Economics and Statistics, 100(3) 550–566, 2018.
5. Orphanides A and Van Norden S.: The unreliability of output-gap estimates in real time The Review of Economics and Statistics, 84(4) 569–583, 2002.
6. Watson M.: Univariate detrending methods with stochastic trends Journal of Monetary Economics, 18(1) 49–75, 1986.

1. Thanks, Ben and Davaa. This is very helpful in showing how to use the BN filter in EViews.

The comparison with the CBO gap is interesting. But I decided to make a direct comparison between the BN filter output gap (with dynamic demeaning) and the HP filter output gap using the EViews procedures, including the add-in.

The result is here.

As can be seen in the linked figure, the BN and HP gaps have more similar amplitude than with the CBO gap. The HP gap is deeper than the BN gap in the 81-82 recession, while the BN gap is deeper than the HP gap in the Great Recession.

Note that I have used an old version of the EViews add-in for which the BN filter needs to be applied to the *first difference* of the series to be detrended, while the HP filter procedure applies directly to the series to be detrended. The blog post seems to suggest that the BN filter can be applied to directly to the series to be detrended, which makes sense since the BN decomposition is a trend/cycle decomposition of y, not dy. So hopefully an updated version of the add-in will allow this. Also, it would be great if users can impose a smoothing parameter delta, much like imposing a lambda for the HP filter. This is possible in the Matlab and R code for the BN filter available on the Global Perspective's Model Hub.

1. Hi James. I just wanted to give you a heads up that both links for the Matlab and R code for the BN filter on Global Perspective's Model Hub currently do not appear to be working. It redirects to Ben's page but gives a 404 error. I just wanted to let you know.

2. Thanks, Yamin. It seems that Ben has reorganized his site. The code is available on his research page.

2. Probably, something is wrong with this add-in. Instead of cycle, it gives trending line. So I cannot replicate this stuff in Eviews 11 and 10. Additionally, Eviews freezes while executing bnfilter_blog.prg.

1. Thanks for bringing it to our attention. We've updated the system with the correct version. Things should work as expected now.

3. Just tried out the updated version of the add-in (version 2.1) and it works perfectly. Once installed, it can be applied easily by opening a series to be detrended (e.g., y, where "genr y=100*log(realGDP)"), clicking the Proc button, and selecting BN Filter under Add-ins at the bottom of the menu.

There is also now an option to impose a smoothing parameter, if desired.

Thanks, Davaa for creating the add-in and updating it to work like the HP filter procedure in EViews, while, of course, producing a more reliable estimate of output gap!

4. How do you set the smoothing delta in command? I tried the code below but no luck:

y.bnfilter(lag=12, demean=1, window=40, delta=0.1)

1. y.bnfilter(lag=12, impose=1, delta=0.1)

2. Thanks. My post below is not relevant after this explanation.

3. Ah I needed to add "impose=1" - thanks for your help!

5. It seems that delta is setting only in add-in now. In command line this is not working at this point.

1. y.bnfilter(lag=12, impose=1, delta=0.1)

6. Just a note that version 2.1 of the add-in, which applies the BN filter to the levels of the series to be detrended, shifts the timing of the cycle one period back. That is, it suggests there is a cycle in the first period that the levels series is available and does not report a cycle in the last period. However, the cycle should be shifted forward one period (i.e., no cycle in the first period that the series is available because the series needs to be differenced when applying the filter and a cycle in the last period). Hopefully this can be fixed in the next version of the add-in. But, in the meantime, it is important for users to shift the cycle forward (e.g., "genr bn_cycle_s_true=bn_cycle_s(-1)") to be correctly timed.

1. This has now been fixed, so the timing is correct for version 2.2 of the add-in. When the BN filter is applied to a series like log real GDP from, say, 1947Q1 to 2020Q2, it will produce an output gap for 1947Q2 to 2020Q2. Also, the default option in version 2.2 is now to apply the dynamic mean adjustment, which is useful for many time series that have changes in long-run drift.

7. Thank you Prof. James for pointing the shift. It will be fixed soon.

1. Thanks, Davaa. And thanks for all the work on the add-in.

2. It has now been fixed. Thanks, again, Davaa.

8. Dear Prof James Morley,

Hopes you are well. I have some problem with the BN Filter add in using Eviews9. After i use this BN filter dialog, then i hit OK. There is error message "@CVAR is an illegal or reserved name".Btw, after i hit OK for the error message, my bn_cycle only show one row1,column1 with value 0. There are also dy_m,dy_s,delta,lag,s1,s2,window,structural break. My question, where is the cycle? What is mean by dy_m?

Hope you can help me

9. Hello! I am the new reader of your blog. I really appreciate the work you do. Your analysis is very professional and interesting. Thanks!

10. Dear Benjamin and Davaajargal great post, thank you. How easy is working these techniques in this software. Best