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Estimating Klein's Model I

klein.Rmd

Equations of Klein’s Model

Stochastic equations

consumptionCt=β0+α1Pt+α2Pt−1+α3(Wtp+Wtg)+ε1t,investmentIt=β0+β1Pt+β2Pt−1+β3Kt−1+ε2t,private wagesWtp=γ0+γ1Yt+γ2Yt−1+γ3At+ε3t. \begin{aligned} \text{consumption}\quad C_t &= \beta_0 + \alpha_1 P_t + \alpha_2 P_{t-1} + \alpha_3\bigl(W^p_t + W^g_t\bigr) + \varepsilon_{1t},\\ \text{investment}\quad I_t &= \beta_0 + \beta_1 P_t + \beta_2 P_{t-1} + \beta_3 K_{t-1} + \varepsilon_{2t},\\ \text{private wages}\quad W^p_t &= \gamma_0 + \gamma_1 Y_t + \gamma_2 Y_{t-1} + \gamma_3 A_t + \varepsilon_{3t}. \end{aligned}

The α\alpha’s, β\beta’s, and γ\gamma’s are the coefficients of the model. PtP_t is the private profits, WtpW^p_t is the private wages, WtgW^g_t is the government wages, Kt−1K_{t-1} is the capital stock at time t−1t-1, and AtA_t is a time trend. The εit\varepsilon_{it} are stochastic error terms.

Equilibrium condition

equilibrium demandYt=Ct+It+Gt \begin{aligned} \text{equilibrium demand}\quad Y_t &= C_t + I_t + G_t \end{aligned}

YtY_t is the equilibrium demand, CtC_t is the consumption, ItI_t is the investment, and GtG_t is the government spending.

Identities

equilibrium demandYt=Ct+It+Gtprivate profitsPt=Yt−Tt−Wtpcapital stockKt=Kt−1+It \begin{aligned} \text{equilibrium demand}\quad Y_t &= C_t + I_t + G_t\\ \text{private profits}\quad P_t &= Y_t - T_t - W^p_t\\ \text{capital stock}\quad K_t &= K_{t-1} + I_t \end{aligned}

Here, TtT_t represents taxes, KtK_t denotes the capital stock at time tt, and GtG_t signifies government spending.

Defining Klein’s Model in R

To estimate and forecast this model, we need to specify the equations, the exogenous variables, and the date ranges for estimation and forecasting.

equations <- "consumption ~ profits + profits.L(1) + total_wages,
investment ~ profits + profits.L(1) + capital_stock.L(1),
wages ~ gdp + gdp.L(1) + time_trend,
gdp == (n_consumption/n_gdp)*consumption + (n_investment/n_gdp)*investment + (n_government/n_gdp)*government,
profits == (n_gdp/n_profits)*gdp - (n_taxes/n_profits)*taxes - (n_wages/n_profits)*wages,
capital_stock == 1.001*capital_stock.L(1) + 0.0002*investment,
total_wages == (n_wages/n_total_wages)*wages + (n_government_wages/n_total_wages)*government_wages"

exogenous_variables <- c("government", "government_wages", "taxes", "time_trend")

The equations string contains the model equations, while exogenous_variables lists the exogenous variables. The .L(1) notation indicates a lag of one period. For further details on the syntax, please refer to the equation syntax documentation.

Creating the SEM

The system_of_equations function is used to create a system of equations object, which can then be used for estimation and forecasting.

library(koma)

sys_eq <- system_of_equations(
  equations = equations,
  exogenous_variables = exogenous_variables
)

print(sys_eq)
## 
## ── System of Equations ─────────────────────────────────────────────────────────
##   consumption ~  constant + profits + profits.L(1) + total_wages
##    investment ~  constant + profits + profits.L(1) + capital_stock.L(1)
##         wages ~  constant + gdp + gdp.L(1) + time_trend
##           gdp == (n_consumption/n_gdp) * consumption + (n_investment/n_gdp) * investment + (n_government/n_gdp) * government
##       profits == (n_gdp/n_profits) * gdp-(n_taxes/n_profits) * taxes-(n_wages/n_profits) * wages
## capital_stock == 1.001 * capital_stock.L(1) + 0.0002 * investment
##   total_wages == (n_wages/n_total_wages) * wages + (n_government_wages/n_total_wages) * government_wages

Defing the Estimation and Forecast Dates

Now, we need to specify the dates for estimation and forecasting.

dates <- list(
  estimation = list(start = c(1990, 1), end = c(2020, 4)),
  forecast = list(start = c(2021, 1), end = c(2023, 4)),
  dynamic_weights = list(start = c(1990, 1), end = c(2020, 4))
)

Preparing the Data

We will use the klein dataset, which contains the required variables for the model. This dataset is a list of standard time series (ts) objects, where each object corresponds to a variable in the model. You can explore the dataset by using the ?klein command to view its documentation and structure.

?klein

To prepare the dataset for use with the koma package, we need to convert the ts objects to the ets format using the as_ets function.

The model is estimated in growth rates. The klein dataset is in levels. We use the ets to tell the model that the data is in levels and how to convert it to growth rates. ets is generalized to accept any additional attributes. Below we use this feature to save the series’ value_type (real or nominal).

# Some series in the klein dataset are nominal. We need to deflate them using the GDP deflator.
klein$profits <- klein$n_profits / (klein$d_gdp / 100)
klein$capital_stock <- klein$n_capital_stock / (klein$d_gdp / 100)
klein$wages <- klein$n_wages / (klein$d_gdp / 100)
klein$government_wages <- klein$n_government_wages / (klein$d_gdp / 100)
klein$taxes <- klein$n_taxes / (klein$d_gdp / 100)

real_series <- c(
  "gdp", "consumption", "investment", "profits", "capital_stock",
  "government", "wages", "government_wages", "taxes"
)
real_ts_data <- lapply(real_series, function(x) {
  as_ets(klein[[x]],
    series_type = "level", method = "diff_log", value_type = "real"
  )
})
names(real_ts_data) <- real_series

real_ts_data$total_wages <- real_ts_data$government_wages + real_ts_data$wages

# net exports can be negative, we thus want to use percentage change instead of log difference
real_ts_data$net_exports <- as_ets(klein$net_exports,
  series_type = "level", method = "percentage", value_type = "real"
)

# To compute the weights in the model, we include the nominal series in the dataset as well.
nominal_series <- c(
  "n_gdp", "n_investment", "n_government", "n_government_wages",
  "n_taxes", "n_profits", "n_wages", "n_consumption"
)
nominal_ts_data <- lapply(nominal_series, function(x) {
  as_ets(klein[[x]],
    series_type = "level", method = "diff_log", value_type = "nominal"
  )
})
names(nominal_ts_data) <- nominal_series

nominal_ts_data$n_total_wages <- nominal_ts_data$n_government_wages + nominal_ts_data$n_wages

# Combining the real and nominal series
ts_data <- c(real_ts_data, nominal_ts_data)

# Adding the time trend
ts_data$time_trend <- ets(seq_len(length(klein$taxes)),
  start = stats::start(ts_data$taxes), frequency = 4,
  series_type = "level", method = "none", value_type = NA
)

With the data prepared, we can now proceed to estimate the model. Before forecasting, however, we need to truncate the endogenous variables to ensure they only include data up to the forecast start date.

ts_data[sys_eq$endogenous_variables] <-
  lapply(sys_eq$endogenous_variables, function(x) {
    stats::window(ts_data[[x]], end = c(2020, 4))
  })

Estimating the Model

The names of the list elements should match the variable names used in the equations. Now, we can estimate the model using the estimate function.

estimates <- estimate(
  sys_eq,
  ts_data = ts_data,
  dates = dates
)
## 
## ── Gibbs Sampler Settings ──────────────────────────────────────────────────────
## ── System Wide Settings ──
## • Number of draws (`ndraws`): 2000
## • Burn-in ratio (`burnin_ratio`): 0.5
## • Burn-in (`burnin`): 1000
## • Store frequency (`nstore`): 1
## • Number of saved draws (`nsave`): 1000
## • Tau (`tau`): 1.1
## 
## 
## ── Estimation ──────────────────────────────────────────────────────────────────
print(estimates)
## 
## ── Estimates ───────────────────────────────────────────────────────────────────
##   consumption ~  - 0.12  +  0.13 * profits - 0.03 * 0.13 * profits.L(1)  +  1.11 * total_wages
##    investment ~  - 0.86  +  0.39 * profits  +  0.16 * 0.39 * profits.L(1)  +  1.51 * capital_stock.L(1)
##         wages ~  - 0.68  +  0.95 * gdp  +  0.43 * 0.95 * gdp.L(1)  +  0 * time_trend
##           gdp == (n_consumption/n_gdp) * consumption  +  (n_investment/n_gdp) * investment  +  (n_government/n_gdp) * government
##       profits == (n_gdp/n_profits) * gdp - (n_taxes/n_profits) * taxes - (n_wages/n_profits) * wages
## capital_stock == 1.001 * capital_stock.L(1)  +  0.0002 * investment
##   total_wages == (n_wages/n_total_wages) * wages  +  (n_government_wages/n_total_wages) * government_wages
summary(estimates)
## 
## =================================================================
##                     consumption     investment     wages         
## -----------------------------------------------------------------
## constant             -0.12           -0.86          -0.68        
##                     [-0.31;  0.08]  [-1.84; 0.02]  [-1.34; -0.06]
## profits.L(1)         -0.03            0.16                       
##                     [-0.05; -0.00]  [ 0.07; 0.24]                
## profits               0.13            0.39                       
##                     [ 0.10;  0.16]  [ 0.21; 0.57]                
## total_wages           1.11                                       
##                     [ 0.97;  1.25]                               
## capital_stock.L(1)                    1.51                       
##                                     [ 0.41; 2.54]                
## gdp.L(1)                                             0.43        
##                                                    [ 0.30;  0.56]
## time_trend                                           0.00        
##                                                    [-0.00;  0.01]
## gdp                                                  0.95        
##                                                    [ 0.80;  1.10]
## =================================================================
## Posterior mean (90% credible interval: [5.0%,  95.0%])

To see how to run the estimation in parallel, refer to the parallel estimation vignette.

Forecasting

To forecast the model, we can use the forecast function. This function will generate forecasts for the specified date range.

forecasts <- forecast(
  estimates,
  dates = dates
)
## 
## ── Forecast ────────────────────────────────────────────────────────────────────
## ✔ Forecasting completed.
print(forecasts)
##         consumption  investment      wages         gdp   profits capital_stock
## 2021 Q1  -1.2940063 -0.99556028 -0.3387788 -0.82305516 -7.322447      2.503406
## 2021 Q2  -0.2981365  1.91448090 -0.4060502 -0.04343018  0.315740      2.506293
## 2021 Q3  -1.2892002  1.09045377 -0.7286282 -0.73506909 -4.884415      2.509017
## 2021 Q4  -0.9010011  1.68225126 -0.6182451 -0.31225408 -1.282972      2.511862
## 2022 Q1  -0.7045260  2.27464617 -0.3376342 -0.21102473 -1.210574      2.514829
## 2022 Q2  -0.9725958  2.00662704 -0.4284649 -0.35510899 -1.942482      2.517745
## 2022 Q3  -1.6702770  0.55227115 -1.0553909 -0.95145610 -5.411500      2.520374
## 2022 Q4  -2.0561957 -0.17733502 -1.5230350 -1.17774912 -5.912272      2.522858
## 2023 Q1  -2.0613849 -0.13030837 -1.6256864 -1.18724541 -5.599824      2.525355
## 2023 Q2  -1.7606452  0.28464001 -1.4495471 -1.00158969 -4.664612      2.527938
## 2023 Q3  -2.1741368 -0.45391948 -1.6479523 -1.29640757 -6.954730      2.530375
## 2023 Q4  -1.8209727 -0.07385907 -1.5616212 -1.07691071 -5.065443      2.532890
##         total_wages government government_wages      taxes time_trend
## 2021 Q1  -0.3791242  1.2757877       -0.5981850  0.8997407        205
## 2021 Q2  -0.3776446 -1.0791366       -0.2234124  9.1808930        206
## 2021 Q3  -0.4831044 -0.3873293        0.8500016  0.3181641        207
## 2021 Q4  -0.6770865 -0.0707721       -0.9965740  4.6250778        208
## 2022 Q1  -0.4214105 -0.8533318       -0.8762856  3.5974820        209
## 2022 Q2  -0.5764775 -0.3751771       -1.3801329 -1.3602277        210
## 2022 Q3  -0.8201585  0.3882606        0.4570688 -5.3550978        211
## 2022 Q4  -1.1938060  1.3228264        0.5937889 -6.2183336        212
## 2023 Q1  -1.2469188  1.2402222        0.8096532 -6.4787268        213
## 2023 Q2  -1.0767545  0.7228893        0.9473758 -2.6579290        214
## 2023 Q3  -1.1609521  1.3819458        1.4832848 -2.8205388        215
## 2023 Q4  -1.1171900  0.8931133        1.2959118 -2.1251498        216
plot(forecasts, variables = "gdp")