Equations of the Small Macro Model

Stochastic equations

\[ \begin{aligned} \text{Consumption}\quad C_t &= \beta_1 + \beta_{1,2} C_{t-1} + \gamma_{1,7} Y_t + \varepsilon_{1t},\\ \text{Investment}\quad I_t &= \beta_2 + \beta_{2,3} I_{t-1} + \varepsilon_{2t},\\ \text{Exports}\quad X_t &= \beta_3 + \beta_{3,4} X_{t-1} + \beta_{3,7} WGDP_t + \varepsilon_{3t},\\ \text{Imports}\quad M_t &= \beta_4 + \gamma_{4,8} D_t + \beta_{4,5} I_{t-1} + \varepsilon_{4t},\\ \text{Prices}\quad P_t &= \beta_5 + \beta_{5,9} ER_t + \beta_{5,10} OP_t + \beta_{5,6} P_{t-1} + \varepsilon_{5t},\\ \text{Interest Rate}\quad R_t &= \beta_6 + \beta_{6,6} P_{t-1} + \beta_{6,8} R_{GE,t} + \gamma_{6,5} P_t + \varepsilon_{6t}. \end{aligned} \]

Equilibrium condition

\[ \begin{aligned} \text{Equilibrium Ouput}\quad GDP_t &= 0.6*C_t + 0.6*D_t + 0.5*X_t - 0.4*I_t \end{aligned} \]

Identities

\[ \begin{aligned} \text{Domestic Demand}\quad D_t &= 0.6*C_t + 0.4*I_t \end{aligned} \]

Coefficients
- The \(\beta\)’s and \(\gamma\)’s are structural coefficients to be estimated.

Endogenous variables

\(C_t\): Consumption at time \(t\).
\(I_t\): Gross fixed capital formation (investment) at time \(t\).
\(X_t\): Exports at time \(t\).
\(M_t\): Imports at time \(t\).
\(P_t\): Price level at time \(t\).
\(R_t\): Short-term interest rate at time \(t\).
\(D_t\): Domestic demand, defined by \(D_t = 0.6 * C_t + 0.4 * I_t\).
\(GDP_t\): Equilibrium output (demand), defined by \(GDP_t = 0.6 * C_t + 0.6 * D_t + 0.5 * X_t - 0.4 * I_t\).

Exogenous variables

\(WGDP_t\): World GDP at time \(t\).
\(ER_t\): Exchange rate at time \(t\).
\(OP_t\): Oil price at time \(t\).
\(R_{GE,t}\): German interest rate at time \(t\).

Other variables

\(G_t\): Government spending at time \(t\).
\(T_t\): Taxes at time \(t\).
\(\varepsilon_{it}\): Stochastic error term for equation \(i\) at time \(t\).

Defining the SEM 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 ~ gdp + consumption.L(1),
investment ~ investment.L(1),
exports ~ world_gdp + exports.L(1),
imports ~ domestic_demand + imports.L(1),
prices ~ exchange_rate + oil_price + prices.L(1),
interest_rate ~ prices + interest_rate_germany + prices.L(1),
gdp == 0.6*consumption + 0.6*domestic_demand + 0.5*exports - 0.4*imports,
domestic_demand == 0.6*consumption + 0.4*investment"

exogenous_variables <- c("world_gdp", "interest_rate_germany", "exchange_rate", "oil_price")

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 + gdp + consumption.L(1)
##      investment ~  constant + investment.L(1)
##         exports ~  constant + world_gdp + exports.L(1)
##         imports ~  constant + domestic_demand + imports.L(1)
##          prices ~  constant + exchange_rate + oil_price + prices.L(1)
##   interest_rate ~  constant + prices + interest_rate_germany + prices.L(1)
##             gdp == 0.6 * consumption + 0.6 * domestic_demand + 0.5 * exports-0.4 * imports
## domestic_demand == 0.6 * consumption + 0.4 * investment

Defing the Estimation and Forecast Dates

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

dates <- list(
  estimation = list(start = c(1996, 1), end = c(2019, 4)),
  forecast = list(start = c(2023, 1), end = c(2023, 4))
)

Preparing the Data

We will use the small_open_economy 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 ?small_open_economy command to view its documentation and structure.

?small_open_economy

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 small_open_economy 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).

series <- names(small_open_economy)
series <- series[!series %in% c("interest_rate", "interest_rate_germany")]

ts_data <- lapply(series, function(x) {
  as_ets(small_open_economy[[x]],
    series_type = "level", method = "diff_log"
  )
})
names(ts_data) <- series

ts_data$interest_rate <- as_ets(small_open_economy$interest_rate,
  series_type = "rate", method = "none"
)
ts_data$interest_rate_germany <- as_ets(small_open_economy$interest_rate_germany,
  series_type = "rate", method = "none"
)

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(2019, 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 ──────────────────────────────────────────────────────────────────
## 
## ── ⚠ MCMC Acceptance Probability Warnings ──────────────────────────────────────
## • consumption: 61.1%
## 
## ℹ Some acceptance probabilities are outside the recommended range (20%-60%).
## Consider revising the equations, tuning each equation's tau, or adjusting your priors.
print(estimates)
## 
## ── Estimates ───────────────────────────────────────────────────────────────────
##     consumption ~  0.36 - 0.02 * gdp  +  0.11 * consumption.L(1)
##      investment ~  0.45  +  0.22 * investment.L(1)
##         exports ~  - 0.29  +  3.18 * world_gdp - 0.29 * exports.L(1)
##         imports ~  - 0.04  +  2.59 * domestic_demand - 0.1 * imports.L(1)
##          prices ~  0.03  +  0.03 * exchange_rate  +  0.01 * oil_price  +  0.56 * prices.L(1)
##   interest_rate ~  - 0.51  +  0.52 * prices  +  0.57 * interest_rate_germany - 0.22 * prices.L(1)
##             gdp == 0.6 * consumption  +  0.6 * domestic_demand  +  0.5 * exports - 0.4 * imports
## domestic_demand == 0.6 * consumption  +  0.4 * investment

summary(estimates)
## 
## ===============================================================================================================
##                        consumption    investment    exports         imports        prices        interest_rate 
## ---------------------------------------------------------------------------------------------------------------
## constant                 0.36          0.45          -0.29           -0.04          0.03          -0.51        
##                        [ 0.27; 0.45]  [0.19; 0.72]  [-0.88;  0.29]  [-0.52; 0.48]  [0.00; 0.06]  [-0.63; -0.39]
## consumption.L(1)         0.11                                                                                  
##                        [-0.08; 0.30]                                                                           
## gdp                     -0.02                                                                                  
##                        [-0.11; 0.07]                                                                           
## investment.L(1)                        0.22                                                                    
##                                       [0.03; 0.40]                                                             
## exports.L(1)                                         -0.29                                                     
##                                                     [-0.46; -0.13]                                             
## world_gdp                                             3.18                                                     
##                                                     [ 2.33;  4.05]                                             
## imports.L(1)                                                         -0.10                                     
##                                                                     [-0.27; 0.06]                              
## domestic_demand                                                       2.59                                     
##                                                                     [ 1.49; 3.51]                              
## prices.L(1)                                                                         0.56          -0.22        
##                                                                                    [0.46; 0.66]  [-0.65;  0.20]
## exchange_rate                                                                       0.03                       
##                                                                                    [0.02; 0.05]                
## oil_price                                                                           0.01                       
##                                                                                    [0.01; 0.01]                
## interest_rate_germany                                                                              0.57        
##                                                                                                  [ 0.52;  0.62]
## prices                                                                                             0.52        
##                                                                                                  [ 0.07;  1.03]
## ===============================================================================================================
## Posterior mean (90% credible interval: [5.0%, 95.0%])
## Estimation period: 1996 Q1 - 2019 Q4

summary(estimates, variables = "investment")
## 
## =============================
##                  investment  
## -----------------------------
## constant          0.45       
##                  [0.19; 0.72]
## investment.L(1)   0.22       
##                  [0.03; 0.40]
## =============================
## Posterior mean (90% credible interval: [5.0%, 95.0%])
## Estimation period: 1996 Q1 - 2019 Q4

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

Diagnostics

You can visualize MCMC diagnostics for coefficient draws using trace_plot().

if (requireNamespace("ggplot2", quietly = TRUE)) {
  trace_plot(estimates, variables = c("consumption", "investment"))
}

You can also inspect running means (cumulative averages) with running_mean_plot().

if (requireNamespace("ggplot2", quietly = TRUE)) {
  running_mean_plot(estimates, variables = c("consumption", "investment"))
}

You can inspect autocorrelation diagnostics with acf_plot().

if (requireNamespace("ggplot2", quietly = TRUE)) {
  acf_plot(estimates, variables = c("consumption", "investment"))
}

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 ────────────────────────────────────────────────────────────────────
## 
## Conditional fill detected after "2019 Q4".
## Missing observations for: "consumption", "investment", "exports", "imports",
## "prices", "interest_rate", "gdp", and "domestic_demand"
## Missing values will be conditionally filled up to "2022 Q4" before forecasting.
## ✔ Forecasting completed.
print(forecasts)
## <koma_ts>
## attributes:
##   series_type: list[12]
##   method: list[12]
##   anker: list[12]
## 
## series:
##         consumption investment exports imports  prices interest_rate    gdp
## 2023 Q1      0.3785     0.4633  1.2160  0.8704 -0.0603        0.8378 0.7344
## 2023 Q2      0.3993     0.5732  0.6695  1.1069 -0.0709        1.2625 0.4129
## 2023 Q3      0.3753     0.6284  0.8870  1.0234  0.0126        1.5708 0.5453
## 2023 Q4      0.3873     0.5874  0.3354  1.0933 -0.0132        1.6856 0.2432
##         domestic_demand world_gdp interest_rate_germany exchange_rate oil_price
## 2023 Q1          0.4124    0.4827                2.3887        0.9217   -8.7520
## 2023 Q2          0.4688    0.4083                3.1460       -1.3863   -3.3787
## 2023 Q3          0.4766    0.4259                3.6393       -1.7869    9.9001
## 2023 Q4          0.4673    0.2906                3.8850       -0.7502   -3.3373

rate(forecasts$mean$gdp)
## <koma_ts>
## attributes:
##   series_type:  chr "rate"
##   method:  chr "diff_log"
##   anker:  num [1:2] 192523 2023
## 
## series:
##           Qtr1      Qtr2      Qtr3      Qtr4
## 2023 0.7343906 0.4128768 0.5453014 0.2431651
level(forecasts$mean$gdp)
## <koma_ts>
## attributes:
##   series_type:  chr "level"
##   method:  chr "diff_log"
## 
## series:
##          Qtr1     Qtr2     Qtr3     Qtr4
## 2022                            192522.6
## 2023 193941.7 194744.1 195808.9 196285.7

Conditional Forecasting

Conditional forecasting allows you to impose restrictions on certain variables during the forecast period to fix their values. This is useful for scenario analysis and policy evaluation, as the forecasts will reflect these user-defined conditions.

fig <- plot(forecasts, variables = "prices")
fig

restrictions <- list(prices = list(value = 0.5, horizon = 1))

forecasts <- forecast(estimates,
  dates = dates,
  restrictions = restrictions
)
plot(forecasts, fig = fig, variables = "prices")
forecasts$mean$prices

Estimating Small Macro Model for Switzerland