Sovereign, getting started

1.1 VAR Covariates

To create our small macro-system we need to first pull four variables from FRED using the quantmod package:

1.Real Gross Domestic Product
2.Real Potential Gross Domestic Product
3.Gross Domestic Product: Implicit Price Deflator
4.Effective Federal Funds Rate

# pull data from FRED 
quantmod::getSymbols.FRED(c('GDPC1','GDPPOT','GDPDEF','FEDFUNDS'), env = globalenv())

## [1] "GDPC1"    "GDPPOT"   "GDPDEF"   "FEDFUNDS"

We then use these four variables to create the quarterly output gap, inflation, and federal funds rate.

# real output gap and inflation
Y = 
  purrr::reduce(
    list(
      data.frame(GDPC1, date = zoo::index(GDPC1)),
      data.frame(GDPPOT, date = zoo::index(GDPPOT)),
      data.frame(GDPDEF, date = zoo::index(GDPDEF))
    ),
    dplyr::inner_join, by = 'date'
  ) %>% 
  dplyr::mutate(
    y = 100*((GDPC1-GDPPOT)/GDPPOT),
    p = 400*(GDPDEF - dplyr::lag(GDPDEF))/dplyr::lag(GDPDEF)) %>%
  dplyr::select(date, y,p)

# federal funds rate
I = data.frame(FEDFUNDS, date = zoo::index(FEDFUNDS)) %>%
  dplyr::group_by(y = lubridate::year(date), q = lubridate::quarter(date)) %>%
  dplyr::summarize(
    date = min(date),
    ff = mean(FEDFUNDS, na.rm = T)) %>%
  dplyr::ungroup() %>%
  dplyr::select(date, ff) %>%
  data.frame()
  
# merge data
Data = 
  dplyr::inner_join(Y, I, by = 'date') %>%
  dplyr::arrange(date) %>%
  dplyr::filter(lubridate::year(date) < 2020) %>%
  na.omit()

head(Data)

##         date           y         p        ff
## 1 1954-07-01 -1.37297039 0.5224804 1.0300000
## 2 1954-10-01 -0.07539062 1.0985239 0.9866667
## 3 1955-01-01  2.15595798 1.8623759 1.3433333
## 4 1955-04-01  3.15032505 1.6629183 1.5000000
## 5 1955-07-01  3.88414881 2.7962536 1.9400000
## 6 1955-10-01  3.82834356 3.9630653 2.3566667

Here one will notice that we have produced a Data.frame with a column named date. This specific data construction is one of four data types accepted by sovereign functions. The other three accepted data types include ts, xts, and zoo time series objects.

1.2. Regimes

Now that we have created out three covariates of interest, we need to create our desired economic regimes. First, we will use an exogenously determined regime, NBER-dated recessions and expansions.

# pull NBER recessions from FRED 
quantmod::getSymbols.FRED(c('USRECQ'), env = globalenv())

## [1] "USRECQ"

# merge recessions into Data 
Data.rec = Data %>%
  dplyr::left_join(
    data.frame(USRECQ, date = zoo::index(USRECQ)),
    by = 'date') %>%
  dplyr::rename(cycle_position = USRECQ) %>%
  dplyr::mutate(cycle_position = dplyr::case_when(cycle_position == 0 ~ 'expansion',
                                                  cycle_position == 1 ~ 'recession'))

head(Data.rec)

##         date           y         p        ff cycle_position
## 1 1954-07-01 -1.37297039 0.5224804 1.0300000      expansion
## 2 1954-10-01 -0.07539062 1.0985239 0.9866667      expansion
## 3 1955-01-01  2.15595798 1.8623759 1.3433333      expansion
## 4 1955-04-01  3.15032505 1.6629183 1.5000000      expansion
## 5 1955-07-01  3.88414881 2.7962536 1.9400000      expansion
## 6 1955-10-01  3.82834356 3.9630653 2.3566667      expansion

While a user may name their regime column anything they would like, for exposition we explicitly name our regime column, cycle_position (referring to the business cycle), and gave clear names to our economic states. We can do this because sovereign will ask us to name declare which column designates our regimes and will then assume that each unique value (numeric or string) in that column is a unique regime.

Second, we will assign two regimes via the k-means clustering algorithm.

# let sovereign determine economic states via k-means clustering
Data.kmeans = 
  sovereign::regimes(
    data = Data,
    regime.n = 2,
    method = 'kmeans')

head(Data.kmeans)

##         date           y         p        ff regime
## 1 1954-07-01 -1.37297039 0.5224804 1.0300000      2
## 2 1954-10-01 -0.07539062 1.0985239 0.9866667      2
## 3 1955-01-01  2.15595798 1.8623759 1.3433333      2
## 4 1955-04-01  3.15032505 1.6629183 1.5000000      2
## 5 1955-07-01  3.88414881 2.7962536 1.9400000      2
## 6 1955-10-01  3.82834356 3.9630653 2.3566667      2

Here we have used sovereign to classify each observation by using the k-means clustering algorithm. One will note that the output of the regime function is the input matrix, Data, concatenated with a new regime column with two unique numeric values (i.e. two mutually exclusive regimes).

We now have our small set of macroeconomic variables, as well as two different sets of economic regimes to introduce state-dependence.

2.1 Single-regime VAR

First we estimate the standard VAR over our macro-system and then visualize the resulting one-step ahead forecasts (left panel) and errors (right panel).

# estimate VAR
var =
    sovereign::VAR(
        data = Data,
        p = 1,
        horizon = 10,
        freq = 'quarter')

# plot forecasting output, one-period ahead horizon
# plot forecasts
sovereign::plot_forecast(var$forecasts[['H_1']])
# plot residuals
sovereign::plot_error(var$residuals[['H_1']])

Second we estimate the associate impulse response functions, bootstrapped confidence intervals, and plot them. Additionally, here we will note that all sovereign plots are based on ggplot2, and the underlying plot objects are the return values for any sovereign plotting function.

# estimate IRF
var.irf =
    sovereign::var_irf(
        var,
        bootstrap.num = 10,
        CI = c(0.05,0.95))

# plot IRF
sovereign::plot_irf(var.irf)

Second, we estimate the forecast error variance decomposition and plot the system of decomposition.

# estimate forecast error variance decomposition
var.fevd =
    sovereign::var_fevd(
        var,
        horizon = 10)

# plot FEVD
sovereign::plot_fevd(var.fevd)

Lastly, we visualize the error variance through time with the historical decomposition.

# estimate historical decomposition
var.hd = sovereign::var_hd(var)

# plot HD
sovereign::plot_hd(var.hd)

2.2 Multi-regime VAR

We next mirror the analysis conducted with the single-regime VAR, but acknowledging the two different phases to the business cycle: expansions and recessions.

First we estimate the regime-dependent VAR, resulting in an expectation function with different parameters based on the conditioning state. That is, we will estimate one VAR per regime, and package these discrete models as one RVAR object - taking the type of a list.

# estimate multi-regime VAR
rvar =
    sovereign::RVAR(
        data = Data.rec,
        regime = 'cycle_position',
        p = 1,
        horizon = 10,
        freq = 'quarter')

# plot forecasting output, one-period ahead horizon
# plot forecasts (right panel)
sovereign::plot_forecast(rvar$forecasts[['H_1']])
# plot residuals (left panel)
sovereign::plot_error(rvar$residuals[['H_1']])

We then estimate impulse response functions to describe the dynamics of the system within each economic state,

# estimate IRF
rvar.irf =
    sovereign::rvar_irf(
        rvar,
        horizon = 10,
        bootstrap.num = 10,
        CI = c(0.05,0.95))

# plot IRF
# regime 1: expansion (right panel)
sovereign::plot_irf(rvar.irf[['regime_expansion']])
# regime 2: recession (left panel)
sovereign::plot_irf(rvar.irf[['regime_recession']])

estimate and visualize the forecast error variance decomposition by state,

# estimate forecast error variance decomposition
rvar.fevd =
    sovereign::rvar_fevd(
        rvar,
        horizon = 10)

# plot FEVD
# regime 1: expansion (right panel)
sovereign::plot_fevd(rvar.fevd[['regime_expansion']])
# regime 2: recession rates (left panel)
sovereign::plot_fevd(rvar.fevd[['regime_recession']])

and analyze the historical decomposition of the regime-dependent model.

# estimate historical decomposition
rvar.hd = sovereign::rvar_hd(rvar)

# plot hd
sovereign::plot_hd(rvar.hd)

Then, upon inspection of both the impulse response functions and forecast error variance decomposition, it is clear that the macroeconomic system behaves differently based on economic state. We will next extend the analysis to assess if there is state-dependent differences in the system, when using local projections.

3.1 Single-Regime Local Projection

# estimate single-regime forecasts 
#  (one or multiple horizons may be estimated)
lp = 
    sovereign::LP(
        data = Data,
        p = 1,
        horizon = c(1:10),
        freq = 'quarter')

# plot forecasting output, one-quarter ahead horizon
# plot forecasts (right panel)
sovereign::plot_forecast(lp$forecasts[['H_1']])
# plot residuals (left panel)
sovereign::plot_error(lp$residuals[['H_1']])

# estimate single-regime IRF
lp.irf = sovereign::lp_irf(lp)

# plot IRF
sovereign::plot_irf(lp.irf)

3.2 Multi-Regime Local Projection

We next turn to estimating state-dependent local projections. However, unlike our VAR experiment, we will here use sovereign’s own regime classifications, attained with the k-means clustering algorithm.

Again, we first estimate and visualize the one-period-ahead forecasts and errors.

# estimate multi-regime IRF
rlp = 
    sovereign::RLP(
        data = Data.kmeans,
        regime = 'regime',
        p = 1,
        horizon = c(1:10),
        freq = 'quarter')

# plot forecasting output, one-quarter ahead horizon
# plot forecasts (right panel)
sovereign::plot_forecast(rlp$forecasts[['H_1']])
# plot residuals (left panel)
sovereign::plot_error(rlp$residuals[['H_1']])

And in our last step, we will create the state-dependent impulse response functions.

# estimate multi-regime IRF
rlp.irf = 
  sovereign::rlp_irf(rlp)

# plot IRF
# regime 1: km-eans cluster 0 (right panel)
sovereign::plot_irf(rlp.irf[[1]])
# regime 2: k-means cluster 1 (left panel)
sovereign::plot_irf(rlp.irf[[2]])

Similar to our VAR analysis, we find that our small macro-system displays different behavior within different states. That is, we have successfully modeled a small-scale macroeconomy across multiple regimes and have identified a clear state-dependence within the system.