Aggregation of Indicators

To align high-frequency indicators (\(x_{t}^{(i)}\)) with the target variable’s frequency (\(y_t\)), a transformation or aggregation process is applied. Let \(K\) represent the number of higher-frequency periods within a single lower-frequency period (e.g., \(K=3\) for monthly data aggregated to a quarter). The aggregation step can be represented as:

\[ \bar{x}_{t}^{(i)} = \sum_{k=0}^{K-1} \omega (k) L^{k/K} x_{t}^{(i)} \]

where:

• The lag operator is defined as \(L^{1/3} x_{t}^{(i)} = x_{t-1/3}^{(i)}\)

• \(x_{t}^{(i)}\): The value of the indicator \(i\) of period \(t\).

• \(\bar{x}_{t}^{(i)}\): The aggregated indicator for period \(t\).

• \(\omega (k)\): The weight assigned to the k-th period of the higher frequency data.

In many applications, \(\omega (k)\) is simply an average over the values in the higher frequency periods. Alternative aggregation techniques include taking the last observation (e.g., the last month of the quarter) or applying weighted averages.

The Model Specification

Once the indicators are aligned with the target frequency, the bridge model is typically specified as a linear regression:

\[ y_t = \beta_0 + \sum_{ i } \sum_{p_i=0}^{P^{(i)}} \beta_{p_i} L^p \bar{x}_{t}^{(i)} + \varepsilon_t \]

where:

• \(P^{(i)}\): The lags to include for indicator \(i\).

• \(\beta_{p_i}\): Coefficients capturing the relationship between the indicators and the target variable.

• \(\varepsilon_t\): The error term.

Bridge models also handle cases where some high-frequency indicators are not fully observed at the time of forecasting. In such cases, missing observations for the current period are imputed or forecast using time series models (e.g., ARIMA, ETS). This allows predictions even when recent observations are missing. This combination of aggregation and forecasting ensures that bridge models are versatile tools for dealing with incomplete data scenarios.

Loading the Data

For this example, the two follwoing Swiss datasets are used:

• gdp: Quarterly GDP data.
• baro: Monthly economic indicator data.

# Load libraries
library(bridgr)
library(tsbox)

# Example data
data("gdp")  # Quarterly GDP data
data("baro") # Monthly economic indicator

gdp <- tsbox::ts_pc(gdp) # Calculate growth rate

# Visualize the data
ts_ggplot(
  ts_scale(ts_c(baro, gdp)),
  title = "Quarterly gdp and monthly economic indicator",
  subtitle = "Scaled to mean 0 and variance 1"
  ) +
  theme_tsbox()

By visualizing the data,it becomes obvious that the monthly economic indicator (baro) is available at a higher frequency than the quarterly GDP data. Moreover, there is a significant correlation.

Estimating the Bridge Model

# Estimate the bridge model
bridge_model <- bridge(
  target = gdp, 
  indic = baro , 
  indic_lags = 1, 
  target_lags=1, 
  h=2 
)
#> The start dates of the target and indicator variables do not match. Aligning them to 2004-04-01
#> Dependent variable: gdp | Frequency: quarter | Estimation sample: 2004-04-01 - 2022-10-01 | Forecast horizon: 2 quarter(s)

Because by calculating the GDP growth rate, there is one observation less at the beginning of the GDP series. The bridgefunction detects mismatched starting dates and aligns them to the earliest common date. The model is then estimated using the specified lags for the target and indicator variables. The h argument specifies the number of periods to forecast the lower frequency variable. The bridge model returns both the dataset the main model was estimated as well as the forecasted dataset for the indicator variables.

# Inspect the datasets
tail(bridge_model$estimation_set)
#> # A tibble: 6 × 4
#>   time        baro baro_lag1   gdp
#>   <date>     <dbl>     <dbl> <dbl>
#> 1 2021-07-01 112.      125.  2.34 
#> 2 2021-10-01 104.      112.  0.411
#> 3 2022-01-01  97.4     104.  0.105
#> 4 2022-04-01  94.3      97.4 1.03 
#> 5 2022-07-01  90.0      94.3 0.255
#> 6 2022-10-01  90.7      90.0 0.102
head(bridge_model$forecast_set)
#> # A tibble: 2 × 3
#>   time        baro baro_lag1
#>   <date>     <dbl>     <dbl>
#> 1 2023-01-01  97.4      90.7
#> 2 2023-04-01  99.8      97.4

Forecasting

# Forecasting using the bridge model
fcst <- forecast(bridge_model)

forecast <- data.frame(
  "time" = fcst$forecast_set$time,
  "forecast" = as.numeric(fcst$mean)
)

# Visualize the forecast
ts_ggplot(
  ts_span(ts_tbl(ts_c(gdp, forecast)), start = 2017),
  title = "Forecasted GDP",
  subtitle = "Bridge model forecast"
) +
  theme_tsbox()

Summary

# Summarize the information in the bridge model
summary(bridge_model)
#> Bridge model summary
#> -----------------------------------
#> Main model:
#> -----------------------------------
#> Series:  gdp 
#> Regression with ARIMA(1,0,0) errors 
#> 
#> Coefficients:
#>          ar1  intercept    baro  baro_lag1
#>       0.2426    -6.4929  0.1615    -0.0917
#> s.e.  0.1213     1.3430  0.0125     0.0121
#> 
#> sigma^2 = 0.5625:  log likelihood = -81.69
#> AIC=173.38   AICc=174.26   BIC=184.9
#> -----------------------------------
#> Single indicator models:
#> -----------------------------------
#> Series:  baro 
#> ARIMA(1,0,2) with non-zero mean 
#> 
#> Coefficients:
#>          ar1     ma1     ma2      mean
#>       0.6688  0.5305  0.3316  100.8580
#> s.e.  0.0653  0.0799  0.0753    1.5774
#> 
#> sigma^2 = 18.46:  log likelihood = -646.14
#> AIC=1302.28   AICc=1302.55   BIC=1319.36
#> Aggregation to low frequency:
#> Using mean over values in corresponding periods.
#> -----------------------------------