Econometrics Toolbox: Getting started

Estimate your first ARIMA model, switch between dynamic econometric specifications, interpret VAR systems and unit root tests, and configure forecast horizons.

← Overview · Getting started · Modeling guide · Methods guide · Examples · Reference

What you will learn

This guide walks you through using the Econometrics Toolbox using the supplied workbook. By the end of this tutorial, you will be able to:

  • Navigate the workbook’s model definition, time series input, and published output sheets;
  • Estimate the baseline US Real GDP Growth ARIMA(1,0,1) model and interpret its maximum likelihood results;
  • Switch models without editing Python to analyze multi-equation Vector Autoregressions (treasury_rates_var);
  • Evaluate stationarity using the dual Augmented Dickey-Fuller (ADF) and KPSS diagnostic workbench (macro_stationarity);
  • Adjust out-of-sample forecast horizons and confidence intervals;
  • Modify autoregressive and moving average lag orders on the Specification sheet to test alternative specifications.

Workbook tour

The Econometrics Toolbox is structured into distinct input and definition sheets and review sheets.

Input Sheets                       Review Sheets
┌───────────────────────┐          ┌───────────────────────┐
│ Start                 │          │ Results               │
│ - Active model        │          │ - Log-Likelihood      │
│ - Forecast horizon    │          │ - AIC / BIC / HQIC    │
│ - Confidence level    │          │ - Fitted Coefficients │
│ - Frequency           │          └───────────────────────┘
├───────────────────────┤          ┌───────────────────────┐
│ Time Series Data      │          │ Forecasts             │
│ - Dates / Periods     │   ────►  │ - Point Projections   │
│ - Series1, 2, 3       │          │ - Lower & Upper CI    │
├───────────────────────┤          │ - Standard Errors     │
│ Specification         │          └───────────────────────┘
│ - Lag orders: p, d, q │          ┌───────────────────────┐
│ - Seasonal P, D, Q, s │          │ Diagnostics           │
│ - VAR & ECM lags      │          │ - ADF / KPSS Unit Root│
└───────────────────────┘          │ - Ljung-Box / ARCH /JB│
                                   └───────────────────────┘

Input and definition sheets

  • Start — Holds high-level run controls in A5:B9: the active model identifier, forecast horizon (1 to 12 steps), confidence level (e.g. 0.95), and reporting frequency. Range D5:H11 lists the model catalog.
  • Time Series Data — Stores date-sequenced historical observations in A5:F200. Columns include Model, Period, Date, Series1, Series2, and Series3.
  • Specification — Configures mathematical orders and lag parameters in A5:D30 for each model family (e.g., \(p, d, q\) for ARIMA, \(P, D, Q, s\) for SARIMAX, and \(\text{lags}\) for VAR/ECM).

Review sheets

  • Results — Populated automatically by the notebook. Displays estimation metrics (summary in A5:B17) including Log-Likelihood, AIC, BIC, HQIC, and sample size, alongside the fitted parameter table (coefficients in D5:I20) detailing coefficients, standard errors, \(z\)-statistics, \(p\)-values, and significance stars.
  • Forecasts — Contains out-of-sample projections (forecasts in A5:F25) showing horizon step numbers, calendar period labels, point estimates, lower/upper confidence bounds, and analytical forecast standard errors.
  • Diagnostics — Reports the 7-step econometric battery (diagnostics in A5:C25) verifying data continuity, convergence status, ADF unit roots, KPSS stationarity, Ljung-Box residual autocorrelation, Jarque-Bera normality, and Engle’s ARCH LM test.

First solve: US Real GDP Growth (ARIMA)

The default active model in the workbook is us_gdp_growth.

1. Inspect the historical time series

Open the Time Series Data sheet and locate the rows for us_gdp_growth. You will find 28 quarterly observations spanning from 2018-Q1 through 2024-Q4. Series1 represents the annualized quarterly percentage growth rate of US Real GDP, including the sharp contraction in 2020-Q2 (-28.0%) and the subsequent rebound in 2020-Q3 (+34.8%).

2. Inspect the model specification

Open the Specification sheet and locate us_gdp_growth. The specification parameters are:

Parameter Value Description
p 1 Autoregressive lag order \(AR(1)\)
d 0 Integration differencing order (growth rates are already stationary)
q 1 Moving average lag order \(MA(1)\)

3. Review estimation results

Open the Results sheet. The notebook has compiled and estimated the ARIMA(1,0,1) model via exact Gaussian maximum likelihood:

  • Status: OPTIMAL
  • Sample periods: 28
  • Log-Likelihood: -95.9986
  • Akaike Information Criterion (AIC): 199.9972
  • Bayesian Information Criterion (BIC): 205.3260
  • Fitted Coefficients:
    • const: 2.833588 (the unconditional mean annualized growth rate)
    • ar.L1: Autoregressive persistence parameter
    • ma.L1: Moving average shock parameter
    • sigma2: Residual innovation variance

4. Review forecasts and prediction intervals

Switch to the Forecasts sheet. Because Forecast horizon is set to 4 on the Start sheet, the model projects forward four quarters:

Horizon Step Period Date Point Forecast Lower CI (95%) Upper CI (95%) Forecast StdError
1 2025-Q1 2.8999 -11.2803 17.0801 7.2349
2 2025-Q2 2.8524 -11.4428 17.1476 7.2936
3 2025-Q3 2.8390 -11.4589 17.1369 7.2950
4 2025-Q4 2.8351 -11.4628 17.1331 7.2950

Notice how the point forecast mean-reverts toward the long-run drift of approximately 2.83%, while the forecast standard error expands as the horizon extends further into the future.

5. Validate econometric diagnostics

Switch to the Diagnostics sheet. The automated diagnostic suite reports:

  • Data continuity: PASS — All 28 quarterly observations are contiguous.
  • Convergence status: PASS — Exact MLE optimization converged successfully.
  • ADF unit root test: PASS — Rejects the unit root null hypothesis (\(p < 0.05\)); the growth series is confirmed stationary.
  • KPSS stationarity test: PASS — Fails to reject stationarity (\(p \ge 0.05\)).
  • Ljung-Box autocorrelation: PASS — Residuals show no significant serial correlation (\(p \ge 0.05\)).
  • Jarque-Bera normality: PASS — Residual distribution is consistent with Gaussian innovations.
  • ARCH heteroskedasticity: PASS — No conditional autoregressive variance clusters detected.

Switching models: Treasury Yield Curve (VAR)

The workbook includes a multivariate Vector Autoregression model linking short-term and long-term interest rates.

1. Change the active model

  1. Navigate to the Start sheet.
  2. Select cell B6 (Active model).
  3. Use the Excel dropdown menu to select treasury_rates_var.

2. Observe the automatic recalculation

The notebook immediately re-evaluates:

  • Reads the paired time series from Time Series Data: Series1 (3-month Treasury yield, ShortRate) and Series2 (10-year Treasury yield, LongRate).
  • Reads lags = 1 from Specification.
  • Estimates a bivariate VAR(1) system across both equations simultaneously.

3. Interpret the multi-equation results

Open the Results sheet. Under Fitted coefficients, you will observe parameter estimates for both equations:

  • ShortRate Equation:
    • ShortRate: const: -0.714262
    • ShortRate: L1.ShortRate: 0.615835 (strong autoregressive persistence)
    • ShortRate: L1.LongRate: cross-rate interaction term
  • LongRate Equation:
    • LongRate: const: constant term
    • LongRate: L1.ShortRate: feedback from short rates to long yields
    • LongRate: L1.LongRate: persistent yield curve structure

In cell C7 of the Diagnostics sheet, review the Granger causality test result: > VAR(1) estimation converged. Granger causality (LongRate -> ShortRate): F=..., p=...

This statistic tests whether historical long-term yields provide statistically significant predictive power for future short-term rate adjustments beyond the short rate’s own history.

Testing stationarity: Macro Stationarity Workbench

A central principle of time series econometrics is distinguishing between non-stationary unit root series (\(I(1)\)) and stationary series (\(I(0)\)).

1. Select the stationarity model

  1. On the Start sheet, set cell B6 (Active model) to macro_stationarity.
  2. Inspect the Results sheet: the status displays NON_STATIONARY.

2. Compare Level vs. Difference series

Open Diagnostics and review the ADF unit root test and KPSS stationarity test rows:

  • Series 1 (Level):
    • ADF \(t\)-statistic: 0.2089 (\(p = 0.9728\)). Because \(p > 0.05\), the test fails to reject the unit root null hypothesis.
    • KPSS statistic: 0.4965 (\(p < 0.05\)). Rejects stationarity.
    • Conclusion: Series 1 contains a stochastic trend (unit root); running OLS regression directly on this series risks spurious correlation.
  • Series 2 (First Difference / Growth Rate):
    • ADF \(t\)-statistic: -5.1731 (\(p = 0.0000\)). Rejects unit root.
    • KPSS statistic: fails to reject stationarity (\(p \ge 0.05\)).
    • Conclusion: First differencing eliminates the stochastic trend, rendering the series stationary and suitable for linear ARMA modeling.

[!TIP] Always pair ADF with KPSS. ADF tests the null hypothesis that a unit root exists, whereas KPSS tests the null hypothesis that the series is stationary. When ADF fails to reject and KPSS rejects, you have definitive confirmatory evidence of a non-stationary unit root process.

Adjusting forecast controls

You can customize forecast projections directly from the spreadsheet without modifying any Python code.

Changing the forecast horizon

  1. On the Start sheet, select cell B7 (Forecast horizon).
  2. Enter 8 (to generate an 8-period forecast).
  3. Switch to the Forecasts sheet. Notice that rows 6 through 13 now present predictions spanning eight full steps (e.g. through 2026-Q4).

Changing the confidence level

  1. On the Start sheet, select cell B8 (Confidence level).
  2. Enter 0.90 (to generate 90% confidence bands) or type 90%.
  3. Switch to the Forecasts sheet:
    • The lower and upper bounds narrow because the critical value \(z_{1 - \alpha/2}\) drops from \(1.960\) (for 95%) to \(1.645\) (for 90%).
    • For us_gdp_growth, Step 1 bounds adjust from [-11.2803, 17.0801] to [-9.0005, 14.8003].

Changing reporting frequency

  1. On the Start sheet, select cell B9 (Frequency).
  2. Select Monthly from the data-validation dropdown.
  3. Switch to the Forecasts sheet: date strings now automatically increment using monthly numbering (e.g. 2025-M01, 2025-M02).

Modifying lag orders

Model lag orders can be calibrated directly on the Specification sheet.

Experimenting with an AR(2) specification

  1. On the Start sheet, ensure Active model is set to us_gdp_growth.
  2. Open the Specification sheet.
  3. Locate row 6, where Parameter is p for us_gdp_growth.
  4. Change cell C6 from 1 to 2.
  5. Switch to the Results sheet:
    • The model specification updates to ARIMA(2,0,1).
    • The Fitted coefficients table now includes both ar.L1 and ar.L2.
    • The information criteria update (AIC adjusts to 201.9833). Because the AIC of 201.98 is higher than the ARIMA(1,0,1) AIC of 199.99, information theory confirms that adding a second autoregressive lag does not justify the extra parameter penalty.

Next steps

  • Explore the Modeling guide to learn how to add your own proprietary time series and configure custom models.
  • Review the Methods guide for mathematical derivations of exact MLE, state-space representations, and diagnostic formulas.
  • Examine the Worked examples for detailed economic commentary on all six included models.