# PMM3 for Time Series: AR, MA, ARMA, and ARIMA Models

#### Serhii Zabolotnii

#### 2026-05-29

## Introduction

Classical time series estimation methods (CSS, MLE) assume Gaussian innovations. The PMM2 method addresses asymmetric non-Gaussian innovations, but many real-world time series exhibit symmetric platykurtic errors — lighter tails than the normal distribution. Examples include:

- Bounded measurement errors (sensor readings within a fixed range)
- Uniformly distributed shocks (e.g., rounding errors)
- Truncated normal processes (censored observations)
- Beta-symmetric disturbances (bounded economic indicators)

The PMM3 method (S=3) extends the Polynomial Maximization Method to these cases, using a cubic stochastic polynomial and Newton-Raphson solver based on the score equation \(Z_{1\nu} = \varepsilon_\nu(\kappa - \varepsilon_\nu^2)\).

This vignette demonstrates PMM3 for time series models:

1. AR — Autoregressive models
2. MA — Moving Average models
3. ARMA — Combined AR + MA models
4. ARIMA — Models with differencing for non-stationary series
5. Real-data examples using datasets included in the package
6. Comparison with MLE and PMM2

### When PMM3 vs PMM2 vs OLS?

Innovation property | Recommended method
--- | ---
Skewed (\(|\gamma_3| > 0.3\)) | PMM2 (`ar_pmm2`, `ma_pmm2`, …)
Symmetric, platykurtic (\(\gamma_4 < -0.7\)) | PMM3 (`ar_pmm3`, `ma_pmm3`, …)
Near-Gaussian | MLE / CSS

Use `pmm_dispatch()` on residuals from a classical fit to automate this choice.

## Setup

    library(EstemPMM)
    set.seed(42)

## Part 1: AR Models with PMM3

### AR(1) with Uniform Innovations

Uniform innovations are the canonical platykurtic case: \(\gamma_4 \approx -1.2\), symmetric, bounded.

    n <- 300
    phi_true <- 0.7

    # Uniform innovations: symmetric, platykurtic (gamma4 ≈ -1.2)
    innov <- runif(n, -sqrt(3), sqrt(3))  # variance = 1

    # Generate AR(1) series
    ar_data <- arima.sim(n = n, list(ar = phi_true), innov = innov)

    # Plot
    plot(ar_data, main = "AR(1) Process with Uniform Innovations",
         ylab = "Value", col = "steelblue", lwd = 1.2)

### Fit AR(1): PMM3 vs MLE

    # PMM3 fit
    fit_pmm3_ar1 <- ar_pmm3(ar_data, order = 1, include.mean = FALSE)

    # MLE fit (classical)
    fit_mle_ar1 <- arima(ar_data, order = c(1, 0, 0), include.mean = FALSE)

    # Compare
    cat("True AR(1) coefficient:", phi_true, "\n")
    #> True AR(1) coefficient: 0.7
    cat("MLE estimate:          ", coef(fit_mle_ar1)["ar1"], "\n")
    #> MLE estimate:           0.7431419
    cat("PMM3 estimate:         ", coef(fit_pmm3_ar1)["ar1"], "\n")
    #> PMM3 estimate:          0.7151052

    # PMM3 diagnostics
    summary(fit_pmm3_ar1)
    #> PMM3 time series estimation results (S = 3, symmetric errors)
    #> Model type: AR(1)
    #> Call:
    #> ts_pmm3(x = x, order = order, model_type = "ar", max_iter = max_iter,
    #>     tol = tol, adaptive = adaptive, step_max = step_max, include.mean = include.mean,
    #>     initial = initial, na.action = na.action, verbose = verbose)
    #>
    #> Coefficients:
    #> AR:       ar1
    #> 0.7151052
    #>
    #> Central moments of initial residuals:
    #>   m2 = 1.013623
    #>   m4 = 1.837523
    #>   m6 = 3.978298
    #>
    #> Theoretical characteristics of PMM3 (S = 3):
    #>   gamma4 = -1.211538
    #>   gamma6 = 6.993114
    #>   g3     = 0.297445  (expected ratio Var[PMM3]/Var[OLS])
    #>   kappa  = 1.292902
    #>
    #> Algorithm information:
    #>   Convergence status: TRUE
    #>   Iterations: 3

The `g3` value in the summary shows the theoretical variance reduction factor. For uniform innovations, \(g_3 \approx 0.64\), meaning PMM3 achieves about 36% variance reduction over OLS.

### AR(2) Model

    phi_true <- c(0.5, -0.3)

    # Beta-symmetric innovations: symmetric, platykurtic
    innov2 <- (rbeta(400, 2, 2) - 0.5) * sqrt(12 * 2)  # scaled for var ≈ 1
    ar2_data <- arima.sim(n = 400, list(ar = phi_true), innov = innov2)

    fit_pmm3_ar2 <- ar_pmm3(ar2_data, order = 2, include.mean = FALSE)
    fit_mle_ar2  <- arima(ar2_data, order = c(2, 0, 0), include.mean = FALSE)

    comparison_ar2 <- data.frame(
      Parameter = c("ar1", "ar2"),
      True   = phi_true,
      MLE    = c(coef(fit_mle_ar2)["ar1"], coef(fit_mle_ar2)["ar2"]),
      PMM3   = coef(fit_pmm3_ar2)[c("ar1", "ar2")]
    )

    print(comparison_ar2, row.names = FALSE)
    #>  Parameter True        MLE       PMM3
    #>        ar1  0.5  0.4502433  0.4468475
    #>        ar2 -0.3 -0.3468338 -0.3250832

    cat("\nPMM3 g3 =", fit_pmm3_ar2@g_coefficient,
        "(expected variance ratio PMM3/OLS)\n")
    #>
    #> PMM3 g3 = 0.6997661 (expected variance ratio PMM3/OLS)

## Part 2: MA Models with PMM3

MA models are more challenging because innovations are not directly observable. PMM3 uses CSS residuals as initial estimates and refines them via Newton-Raphson.

### MA(1) Model

    n <- 300
    theta_true <- 0.6

    # Truncated normal innovations (|x| ≤ 2): symmetric, platykurtic
    innov_raw <- rnorm(n * 2)
    innov_tn <- innov_raw[abs(innov_raw) <= 2]
    innov_tn <- innov_tn[1:n]
    innov_tn <- innov_tn / sd(innov_tn)  # standardize

    # Generate MA(1) series
    ma_data <- arima.sim(n = n, list(ma = theta_true), innov = innov_tn)

    # Fit
    fit_pmm3_ma1 <- ma_pmm3(ma_data, order = 1, include.mean = FALSE)
    fit_mle_ma1  <- arima(ma_data, order = c(0, 0, 1), include.mean = FALSE)

    cat("True MA(1) coefficient:", theta_true, "\n")
    #> True MA(1) coefficient: 0.6
    cat("MLE estimate:          ", coef(fit_mle_ma1)["ma1"], "\n")
    #> MLE estimate:           0.6569534
    cat("PMM3 estimate:         ", coef(fit_pmm3_ma1)["ma1"], "\n")
    #> PMM3 estimate:          0.6323384
    cat("\nConvergence:", fit_pmm3_ma1@convergence, "\n")
    #>
    #> Convergence: TRUE
    cat("Iterations: ", fit_pmm3_ma1@iterations, "\n")
    #> Iterations:  3

### MA(2) Model

    theta_true <- c(0.5, 0.3)

    innov_u <- runif(400, -sqrt(3), sqrt(3))
    ma2_data <- arima.sim(n = 400, list(ma = theta_true), innov = innov_u)

    fit_pmm3_ma2 <- ma_pmm3(ma2_data, order = 2, include.mean = FALSE)
    fit_mle_ma2  <- arima(ma2_data, order = c(0, 0, 2), include.mean = FALSE)

    comparison_ma2 <- data.frame(
      Parameter = c("ma1", "ma2"),
      True   = theta_true,
      MLE    = c(coef(fit_mle_ma2)["ma1"], coef(fit_mle_ma2)["ma2"]),
      PMM3   = coef(fit_pmm3_ma2)[c("ma1", "ma2")]
    )

    print(comparison_ma2, row.names = FALSE)
    #>  Parameter True       MLE      PMM3
    #>        ma1  0.5 0.5262435 0.5171439
    #>        ma2  0.3 0.3078768 0.2931746

## Part 3: ARMA Models

ARMA models combine autoregressive and moving average components. PMM3 handles both components simultaneously.

### ARMA(1,1) Model

    n <- 400
    phi_true   <- 0.5
    theta_true <- 0.3

    # Uniform innovations
    innov_arma <- runif(n, -sqrt(3), sqrt(3))

    arma_data <- arima.sim(n = n,
                           list(ar = phi_true, ma = theta_true),
                           innov = innov_arma)

    # Fit
    fit_pmm3_arma <- arma_pmm3(arma_data, order = c(1, 1), include.mean = FALSE)
    fit_mle_arma  <- arima(arma_data, order = c(1, 0, 1), include.mean = FALSE)

    cat("True: AR =", phi_true, ", MA =", theta_true, "\n\n")
    #> True: AR = 0.5 , MA = 0.3

    cat("MLE estimates:\n")
    #> MLE estimates:
    print(coef(fit_mle_arma)[c("ar1", "ma1")])
    #>       ar1       ma1
    #> 0.4258325 0.3115166

    cat("\nPMM3 estimates:\n")
    #>
    #> PMM3 estimates:
    print(coef(fit_pmm3_arma))
    #>       ar1       ma1
    #> 0.4254178 0.3708970

    summary(fit_pmm3_arma)
    #> PMM3 time series estimation results (S = 3, symmetric errors)
    #> Model type: ARMA(1,1)
    #> Call:
    #> ts_pmm3(x = x, order = order, model_type = "arma", max_iter = max_iter,
    #>     tol = tol, adaptive = adaptive, step_max = step_max, include.mean = include.mean,
    #>     initial = initial, na.action = na.action, verbose = verbose)
    #>
    #> Coefficients:
    #> AR:       ar1
    #> 0.4254178
    #> MA:      ma1
    #> 0.370897
    #>
    #> Central moments of initial residuals:
    #>   m2 = 1.007571
    #>   m4 = 1.773758
    #>   m6 = 3.745631
    #>
    #> Theoretical characteristics of PMM3 (S = 3):
    #>   gamma4 = -1.252799
    #>   gamma6 = 7.453816
    #>   g3     = 0.2795877  (expected ratio Var[PMM3]/Var[OLS])
    #>   kappa  = 1.270544
    #>
    #> Algorithm information:
    #>   Convergence status: TRUE
    #>   Iterations: 3

### Diagnostic Plots

    plot(fit_pmm3_arma)

The four diagnostic panels:

1. Residuals vs Time — should show no patterns
2. Q-Q Plot — deviations from the line indicate non-normality (expected for platykurtic errors: lighter tails than normal)
3. ACF — should fall within confidence bands (white noise)
4. Histogram — platykurtic residuals appear flatter than a bell curve

## Part 4: ARIMA Models with Differencing

For non-stationary series, ARIMA models apply differencing before estimation. PMM3 uses a nonlinear solver with `stats::arima(fixed=...)` for proper residual computation and `numDeriv::jacobian` for derivatives. This approach correctly captures the full ARIMA dynamics after differencing.

### ARIMA(1,1,0) Model

    n <- 300
    phi_true <- 0.6

    # Generate stationary AR(1) with uniform innovations
    innov_arima <- runif(n, -sqrt(3), sqrt(3))
    x_stat <- arima.sim(n = n, list(ar = phi_true), innov = innov_arima)

    # Integrate (cumulative sum) to create non-stationary series
    x_integrated <- cumsum(x_stat)

    plot(x_integrated, type = "l",
         main = "ARIMA(1,1,0) Process (Integrated AR(1))",
         ylab = "Value", xlab = "Time", col = "darkgreen", lwd = 1.2)

### Fit ARIMA(1,1,0)

    fit_pmm3_arima <- arima_pmm3(x_integrated, order = c(1, 1, 0),
                                 include.mean = FALSE)
    fit_mle_arima  <- arima(x_integrated, order = c(1, 1, 0),
                            include.mean = FALSE)

    cat("True AR(1) coefficient:", phi_true, "\n")
    #> True AR(1) coefficient: 0.6
    cat("MLE estimate:          ", coef(fit_mle_arima)["ar1"], "\n")
    #> MLE estimate:           0.5788831
    cat("PMM3 estimate:         ", coef(fit_pmm3_arima)["ar1"], "\n")
    #> PMM3 estimate:          0.6318376

    summary(fit_pmm3_arima)
    #> PMM3 time series estimation results (S = 3, symmetric errors)
    #> Model type: ARIMA(1,1,0)
    #> Call:
    #> ts_pmm3(x = x, order = order, model_type = "arima", max_iter = max_iter,
    #>     tol = tol, adaptive = adaptive, step_max = step_max, include.mean = include.mean,
    #>     initial = initial, na.action = na.action, verbose = verbose)
    #>
    #> Coefficients:
    #> AR:       ar1
    #> 0.6318376
    #>
    #> Central moments of initial residuals:
    #>   m2 = 1.034877
    #>   m4 = 1.910823
    #>   m6 = 4.259607
    #>
    #> Theoretical characteristics of PMM3 (S = 3):
    #>   gamma4 = -1.215801
    #>   gamma6 = 7.080315
    #>   g3     = 0.3086522  (expected ratio Var[PMM3]/Var[OLS])
    #>   kappa  = 1.2847
    #>
    #> Algorithm information:
    #>   Convergence status: TRUE
    #>   Iterations: 5

### ARIMA(1,1,1) Model

    phi_true   <- 0.4
    theta_true <- 0.3

    innov_111 <- runif(350, -sqrt(3), sqrt(3))
    x_stat2 <- arima.sim(n = 350, list(ar = phi_true, ma = theta_true),
                         innov = innov_111)
    x_int2 <- cumsum(x_stat2)

    fit_pmm3_111 <- arima_pmm3(x_int2, order = c(1, 1, 1), include.mean = FALSE)
    fit_mle_111  <- arima(x_int2, order = c(1, 1, 1), include.mean = FALSE)

    comparison_arima <- data.frame(
      Parameter = c("ar1", "ma1"),
      True   = c(phi_true, theta_true),
      MLE    = c(coef(fit_mle_111)["ar1"], coef(fit_mle_111)["ma1"]),
      PMM3   = coef(fit_pmm3_111)[c("ar1", "ma1")]
    )

    print(comparison_arima, row.names = FALSE)
    #>  Parameter True       MLE      PMM3
    #>        ar1  0.4 0.3234249 0.3745386
    #>        ma1  0.3 0.2558448 0.3151259

## Part 5: Forecasting with PMM3 Models

All PMM3 time series objects support the `predict()` method.

    # Forecast 10 steps ahead from the AR(1) model
    preds_ar <- predict(fit_pmm3_ar1, n.ahead = 10)

    cat("AR(1) PMM3 forecasts (10 steps):\n")
    #> AR(1) PMM3 forecasts (10 steps):
    print(round(as.numeric(preds_ar), 4))
    #>  [1] -1.8096 -1.2941 -0.9254 -0.6618 -0.4732 -0.3384 -0.2420 -0.1731 -0.1238
    #> [10] -0.0885

    # Plot original series with forecasts
    n_plot <- 80
    plot_data <- tail(as.numeric(ar_data), n_plot)
    pred_vals <- as.numeric(preds_ar)

    plot(seq_along(plot_data), plot_data, type = "l",
         xlim = c(1, n_plot + 10),
         ylim = range(c(plot_data, pred_vals)),
         main = "AR(1) PMM3: Observed and Forecast",
         xlab = "Time", ylab = "Value",
         col = "steelblue", lwd = 1.5)
    lines(n_plot + 1:10, pred_vals, col = "red", lwd = 2, lty = 2)
    legend("topleft", legend = c("Observed", "Forecast"),
           col = c("steelblue", "red"), lty = c(1, 2), lwd = c(1.5, 2))

### Forecasting ARMA Models

    # Forecast from the ARMA(1,1) model
    preds_arma <- predict(fit_pmm3_arma, n.ahead = 10)

    cat("ARMA(1,1) PMM3 forecasts (10 steps):\n")
    #> ARMA(1,1) PMM3 forecasts (10 steps):
    print(round(as.numeric(preds_arma), 4))
    #>  [1] 1.4471 0.6156 0.2619 0.1114 0.0474 0.0202 0.0086 0.0036 0.0016 0.0007

## Part 6: Method Selection with pmm_dispatch()

The `pmm_dispatch()` function automates the choice between OLS, PMM2, and PMM3 by analyzing innovation properties from an initial classical fit.

    # Fit classical model first
    fit_classical <- arima(ar_data, order = c(1, 0, 0), include.mean = FALSE)
    res_classical <- residuals(fit_classical)

    # Dispatch
    recommendation <- pmm_dispatch(res_classical)
    #> PMM method dispatch
    #>   n = 300 | gamma3 = +0.014 | gamma4 = -1.216
    #>   g2(PMM2) = 0.9998 | g3(PMM3) = 0.2948
    #>   Selected: PMM3 (expected variance reduction: 70.5%)
    #>   Reason: |gamma3| = 0.014 < 0.3 (symmetric) and gamma4 = -1.216 < -0.7 (platykurtic). PMM3 expected to reduce variance by 70.5%.

    # Compare all three methods on uniform innovations
    cat("Recommended method:", recommendation$method, "\n")
    #> Recommended method: PMM3
    cat("gamma3 =", round(recommendation$gamma3, 3),
        " gamma4 =", round(recommendation$gamma4, 3), "\n")
    #> gamma3 = 0.014  gamma4 = -1.216

    if (!is.null(recommendation$g3)) {
      cat("g3 =", round(recommendation$g3, 3),
          " (PMM3 variance reduction factor)\n")
    }
    #> g3 = 0.295  (PMM3 variance reduction factor)

## Part 7: Real-Data Example — DJIA 2002

The `djia2002` dataset contains daily closing prices and changes of the Dow Jones Industrial Average for July–December 2002.

    data(djia2002)
    changes <- na.omit(djia2002$change)
    n_obs <- length(changes)

    plot(changes, type = "l",
         main = "DJIA Daily Changes (Jul-Dec 2002)",
         ylab = "Change (USD)", xlab = "Trading Day",
         col = "steelblue", lwd = 1.2)
    abline(h = 0, col = "red", lty = 2)

### Analyze Innovation Properties

    # Fit a classical AR(1)
    fit_djia_mle <- arima(changes, order = c(1, 0, 0))
    res_djia <- residuals(fit_djia_mle)

    cat("Innovation diagnostics:\n")
    #> Innovation diagnostics:
    cat("  Skewness (gamma3):", round(pmm_skewness(res_djia), 3), "\n")
    #>   Skewness (gamma3): 0.445
    cat("  Kurtosis (gamma4):", round(pmm_kurtosis(res_djia), 3), "\n")
    #>   Kurtosis (gamma4): 0.263
    cat("  Symmetry test:\n")
    #>   Symmetry test:
    sym <- test_symmetry(res_djia)
    cat("    symmetric:", sym$is_symmetric, "\n")
    #>     symmetric: FALSE
    cat("    message:  ", sym$message, "\n")
    #>     message:   |gamma3| = 0.445 > 0.3: residuals are asymmetric. Consider PMM2 instead.

    # Let pmm_dispatch decide
    dispatch <- pmm_dispatch(res_djia)
    #> PMM method dispatch
    #>   n = 126 | gamma3 = +0.445 | gamma4 = +0.263
    #>   g2(PMM2) = 0.9126 | g3(PMM3) = 0.9900
    #>   Selected: PMM2 (expected variance reduction: 8.7%)
    #>   Reason: |gamma3| = 0.445 > 0.3 and g2 = 0.913 < 0.95: moderate asymmetry, PMM2 worthwhile (8.7% reduction).
    cat("\nRecommended method:", dispatch$method, "\n")
    #>
    #> Recommended method: PMM2

### Fit AR(1) with All Three Methods

    fit_djia_pmm2 <- ar_pmm2(changes, order = 1)
    fit_djia_pmm3 <- ar_pmm3(changes, order = 1)

    comparison_djia <- data.frame(
      Method = c("MLE", "PMM2", "PMM3"),
      ar1 = c(coef(fit_djia_mle)["ar1"],
              coef(fit_djia_pmm2)["ar1"],
              coef(fit_djia_pmm3)["ar1"])
    )
    print(comparison_djia, row.names = FALSE)
    #>  Method         ar1
    #>     MLE -0.03956444
    #>    PMM2 -0.02144724
    #>    PMM3 -0.04472054

The DJIA data is known to have positive skewness, so `pmm_dispatch()` typically recommends PMM2 as the appropriate method for this series. PMM3 may still converge but is not expected to improve upon PMM2 when the innovations are asymmetric.

## Part 8: Real-Data Example — WTI Crude Oil Prices

The `DCOILWTICO` dataset contains WTI crude oil daily spot prices.

    data(DCOILWTICO)

    # Compute log-returns (remove NAs)
    prices <- na.omit(DCOILWTICO$DCOILWTICO)
    log_returns <- diff(log(prices))
    log_returns <- log_returns[is.finite(log_returns)]

    plot(log_returns, type = "l",
         main = "WTI Crude Oil: Daily Log-Returns",
         ylab = "Log-return", xlab = "Trading Day",
         col = "darkgreen", lwd = 0.8)
    abline(h = 0, col = "red", lty = 2)

### Innovation Analysis

    # Fit AR(1) to log-returns
    fit_oil_mle <- arima(log_returns, order = c(1, 0, 0))
    res_oil <- residuals(fit_oil_mle)

    cat("Log-return innovation diagnostics:\n")
    #> Log-return innovation diagnostics:
    cat("  Skewness (gamma3):", round(pmm_skewness(res_oil), 3), "\n")
    #>   Skewness (gamma3): -0.498
    cat("  Kurtosis (gamma4):", round(pmm_kurtosis(res_oil), 3), "\n")
    #>   Kurtosis (gamma4): 2.093

    # Dispatch
    dispatch_oil <- pmm_dispatch(res_oil)
    #> PMM method dispatch
    #>   n = 1248 | gamma3 = -0.498 | gamma4 = +2.093
    #>   g2(PMM2) = 0.9394 | g3(PMM3) = 0.9010
    #>   Selected: PMM2 (expected variance reduction: 6.1%)
    #>   Reason: |gamma3| = 0.498 > 0.3 and g2 = 0.939 < 0.95: moderate asymmetry, PMM2 worthwhile (6.1% reduction).
    cat("\nRecommended method:", dispatch_oil$method, "\n")
    #>
    #> Recommended method: PMM2

### Fit and Compare

    fit_oil_pmm2 <- ar_pmm2(log_returns, order = 1)
    fit_oil_pmm3 <- ar_pmm3(log_returns, order = 1)

    comparison_oil <- data.frame(
      Method = c("MLE", "PMM2", "PMM3"),
      ar1 = c(coef(fit_oil_mle)["ar1"],
              coef(fit_oil_pmm2)["ar1"],
              coef(fit_oil_pmm3)["ar1"])
    )
    print(comparison_oil, row.names = FALSE)
    #>  Method        ar1
    #>     MLE 0.02971020
    #>    PMM2 0.02250418
    #>    PMM3 0.04288416

    vf2_oil <- pmm2_variance_factor(fit_oil_pmm2@m2, fit_oil_pmm2@m3, fit_oil_pmm2@m4)
    cat("\nPMM2 g2 =", vf2_oil$g2, "\n")
    #>
    #> PMM2 g2 =
    cat("PMM3 g3 =", fit_oil_pmm3@g_coefficient, "\n")
    #> PMM3 g3 = 0.9002976

Crude oil log-returns typically exhibit heavy tails (positive excess kurtosis), so PMM2 is usually the better choice. This example demonstrates how `pmm_dispatch()` guides users to the correct method.

## Part 9: Adaptive Mode

By default, PMM3 computes \(\kappa\) from the initial residuals and holds it fixed. In adaptive mode, \(\kappa\) is re-estimated from the current residuals at each Newton-Raphson iteration:

    # Fixed kappa (default)
    fit_fixed <- ar_pmm3(ar_data, order = 1, include.mean = FALSE, adaptive = FALSE)

    # Adaptive kappa
    fit_adapt <- ar_pmm3(ar_data, order = 1, include.mean = FALSE, adaptive = TRUE)

    comparison_adaptive <- data.frame(
      Mode   = c("Fixed kappa", "Adaptive kappa"),
      ar1    = c(coef(fit_fixed)["ar1"], coef(fit_adapt)["ar1"]),
      g3     = c(fit_fixed@g_coefficient, fit_adapt@g_coefficient),
      iters  = c(fit_fixed@iterations, fit_adapt@iterations)
    )
    print(comparison_adaptive, row.names = FALSE)
    #>            Mode       ar1       g3 iters
    #>     Fixed kappa 0.7151052 0.297445     3
    #>  Adaptive kappa 0.7147911 0.297445     4

Adaptive mode may improve estimates when the distribution departs significantly from the initial OLS residuals, at the cost of additional iterations.

## Part 10: Model Comparison via AIC

The `AIC()` method is available for all PMM3 time series objects.

    # Fit several AR orders with PMM3
    aic_vals <- sapply(1:5, function(p) {
      fit <- ar_pmm3(ar_data, order = p, include.mean = FALSE)
      AIC(fit)
    })

    aic_df <- data.frame(Order = 1:5, AIC = round(aic_vals, 2))
    print(aic_df, row.names = FALSE)
    #>  Order    AIC
    #>      1 858.54
    #>      2 861.11
    #>      3 863.25
    #>      4 865.24
    #>      5 867.37

    cat("\nBest AR order by AIC:", which.min(aic_vals), "\n")
    #>
    #> Best AR order by AIC: 1

## Practical Guidelines

### When to Use PMM3 for Time Series

Recommended when innovations are:

- Symmetric (\(|\gamma_3| \leq 0.3\))
- Platykurtic (\(\gamma_4 < -0.7\))
- Example distributions: uniform, beta-symmetric, truncated normal
- Sample size \(n \geq 200\) for reliable moment estimation

Use PMM2 instead when:

- Innovations are skewed (\(|\gamma_3| > 0.3\))
- Financial returns with asymmetric tails

Stick with MLE/CSS when:

- Innovations are approximately Gaussian
- Very small samples (\(n < 100\))

### Diagnostic Workflow

1. Fit a classical model (MLE/CSS) as baseline
2. Examine residuals for non-normality:
   - `pmm_skewness(residuals)` — check symmetry
   - `pmm_kurtosis(residuals)` — check tail behavior
   - `test_symmetry(residuals)` — formal symmetry check
3. Use `pmm_dispatch(residuals)` for automatic method selection
4. Refit with PMM3 if platykurtic and symmetric
5. Compare AIC across model orders
6. Validate with diagnostic plots and out-of-sample forecasts

### Computational Notes

- AR models: Fast convergence (3–10 iterations), linearized solver
- MA models: Moderate (10–30 iterations), linearized solver
- ARMA models: Moderate (10–30 iterations), linearized solver
- ARIMA models: Nonlinear solver with numerical Jacobian — slower but accurately captures differencing dynamics

## Efficiency Summary

Based on Monte Carlo simulations (R = 1000, n = 200, uniform innovations):

Model | MSE Ratio (PMM3/MLE) | Variance Reduction
--- | --- | ---
AR(1) | 0.55–0.70 | 30–45%
AR(2) | 0.60–0.75 | 25–40%
MA(1) | 0.90–1.00 | 0–10%
ARMA(1,1) | 0.70–0.85 | 15–30%
ARIMA(1,1,0) | 0.32–0.70 | 30–68%

Key insight: PMM3 provides the largest gains for AR and ARIMA models with platykurtic innovations. MA models show smaller improvements because CSS residuals are less precise starting points, but PMM3 remains consistent.

## Available Functions

Function | Model | Class Returned
--- | --- | ---
`ar_pmm3()` | AR(p) | `ARPMM3`
`ma_pmm3()` | MA(q) | `MAPMM3`
`arma_pmm3()` | ARMA(p,q) | `ARMAPMM3`
`arima_pmm3()` | ARIMA(p,d,q) | `ARIMAPMM3`
`ts_pmm3()` | Any of above | `TS3fit` subclass

All classes support: `coef()`, `residuals()`, `fitted()`, `predict()`, `summary()`, `plot()`, `AIC()`.

## Next Steps

- PMM3 for linear regression: See `vignette("pmm3_symmetric_errors")`
- PMM2 time series: See `vignette("pmm2_time_series")` for asymmetric innovations
- Bootstrap inference: See `vignette("bootstrap_inference")`
- PMM2 basics: See `vignette("pmm2_introduction")`

## References

1. Zabolotnii S., Warsza Z.L., Tkachenko O. (2018). Polynomial Estimation of Linear Regression Parameters for the Asymmetric PDF of Errors. Springer AISC, vol 743. doi:10.1007/978-3-319-77179-3_75

2. Zabolotnii S., Tkachenko O., Warsza Z.L. (2022). Application of the Polynomial Maximization Method for Estimation Parameters of Autoregressive Models with Asymmetric Innovations. Springer AISC, vol 1427. doi:10.1007/978-3-031-03502-9_37

3. Package functions: `?ar_pmm3`, `?ma_pmm3`, `?arma_pmm3`, `?arima_pmm3`

4. Method selection: `?pmm_dispatch`, `?test_symmetry`

5. GitHub: https://github.com/SZabolotnii/EstemPMM