Data Ingestion and Temporal Structuring

Quarterly population shift records spanning from September 1971 to June 1993 are imported and converted into a formal time series object. The dataset is structured with a quarterly frequency to align with the observation intervals.

raw_values <- read.table("quarterly_population.dat")$V1
demographic_ts <- ts(raw_values, start = c(1971, 3), frequency = 4)
plot(demographic_ts, ylab = "Population Change", main = "Quarterly Demographic Shifts (1971-1993)")

Visual inspection of the generated plot reveals an absence of deterministic trends or seasonal cycles. The series fluctuates around a constant mean, suggesting potential stationarity. Formal statistical validation is required to confirm this observation.

Stationarity Verification

The Augmented Dickey-Fuller (ADF) test is applied to rigorously evaluate the null hypothesis of a unit root against the alternative of stationarity.

library(tseries)
stationarity_check <- adf.test(demographic_ts, alternative = "stationary")
print(stationarity_check)

The test output yields a p-value below the 0.05 threshold across multiple lag specifications, leading to the rejection of the unit root hypothesis. The series is statistically stationary, permitting direct modeling without differencing.

Serial Correlation Asssessment

Before proceeding to model identification, the series must be evaluated for pure randomness. The Ljung-Box test examines whether autocorrelations up to specified lags differ significantly from zero.

lb_results <- sapply(c(2, 4), function(h) {
  Box.test(demographic_ts, lag = h, type = "Ljung-Box")
})
print(lb_results)

At lag 2, the p-value exceeds 0.05, indicating insufficient evidence against randomness. However, at lag 4, the p-value drops to approximately 0.0026, strongly rejecting the white noise hypothesis. The confirmed presence of serial correlation justifies the extraction of temporal dependencies using an autoregressive moving average framework.

Order Identification via ACF/PACF and AIC Optimization

Autocorrelation (ACF) and partial autocorrelation (PACF) functions are plotted to diagnose the underlying dependency structure.

acf(demographic_ts, main = "Autocorrelation Function")
pacf(demographic_ts, main = "Partial Autocorrelation Function")

Both correlograms exhibit gradual decay rather than sharp cutoffs, indicating a mixed autoregressive and moving average process. To determine the optimal lag orders $(p, q)$, a grid search evaluates ARMA configurasions ranging from (1,1) to (4,4) using Maximum Likelihood estimation. The Akaike Information Criterion (AIC) serves as the selection metric.

param_grid <- expand.grid(p = 1:4, q = 1:4)
aic_scores <- apply(param_grid, 1, function(row) {
  fit <- tryCatch(
    arima(demographic_ts, order = c(row[1], 0, row[2]), method = "ML"),
    error = function(e) NULL
  )
  if (!is.null(fit)) return(fit$aic) else return(NA)
})
param_grid$aic <- aic_scores
optimal_params <- param_grid[which.min(param_grid$aic), ]
print(optimal_params)

The optimization routine identifies the ARMA(3,2) configuration as the most parsimonious model, achieving the minimum AIC value of approximately 768.05.

Parameter Estimation and Multi-Step Projection

The selected ARMA(3,2) structure is fitted to the historical data. Subsequently, a 20-quarter (5-year) forecast is generated, complete with 80% and 95% prediction intervals.

final_model <- arima(demographic_ts, order = c(optimal_params$p, 0, optimal_params$q), method = "ML")

library(forecast)
projection_horizon <- 20
proj <- forecast(final_model, h = projection_horizon, level = c(80, 95))
print(proj)

plot(proj, lty = 2, main = "5-Year Demographic Projection")
lines(fitted(final_model), col = "steelblue", lwd = 2)

The numerical output provides point estimates alongside lower and upper confidence bounds for each future quarter. The visualization overlays the historical fitted values with the forward-looking projection, demonstrating stable long-term behavior and narrowing uncertainty bands as the model converges to the series mean.