Econometric modelling is a branch of economics that employs statistical methods to analyse economic phenomena, forecast future trends, and evaluate economic policies. It combines economic theory, mathematics, and statistical techniques to quantify and understand the relationships between economic variables.
Econometric models go beyond univariate time series methods — they explain why a variable behaves as it does by relating it to other economic variables (its determinants). This gives forecasts a causal interpretation, which is valuable for policy analysis.
Econometric models are used to:
| Application Area | Example |
|---|---|
| Macroeconomic Forecasting | Central banks forecast GDP growth, inflation, unemployment rates, and interest rates using large structural models. |
| Financial Markets | The Capital Asset Pricing Model (CAPM) and Arbitrage Pricing Theory (APT) estimate expected asset returns based on risk. |
| Labour Economics | Researchers use difference-in-differences and regression discontinuity designs to evaluate minimum wage laws. |
| Health Economics | Instrumental variables and propensity score matching address endogeneity in evaluating healthcare interventions. |
| Environmental Economics | Panel data and spatial econometrics assess the effectiveness of environmental regulations on economic growth. |
| Marketing & Consumer Behaviour | Discrete choice models and time series analysis forecast consumer demand and advertising effectiveness. |
The variables used in econometric models are:
The general functional form is:
\[y_t = f(x_{1t},\, x_{2t},\, \ldots,\, x_{mt}) \tag{1}\]
Equation (1) states that the level of \(y_t\) is influenced by the behaviour of variables \(x_{1t}, x_{2t}, \ldots, x_{mt}\), and the relationship is established from historical data.
The standard multiple linear regression model is:
\[y_t = \beta_0 + \beta_1 x_{1t} + \beta_2 x_{2t} + \cdots + \beta_m x_{mt} + \varepsilon_t \tag{2}\]
In compact summation form:
\[y_t = \beta_0 + \sum_{i=1}^{m} \beta_i x_{it} + \varepsilon_t \tag{3}\]
where:
A more general model, the Autoregressive Distributed Lag (ADL) model, allows both lagged values of the dependent variable and lagged values of the independent variables:
\[y_t = \beta_0 + \beta_1 x_{1t} + \cdots + \beta_m x_{mt} + \phi_1 y_{t-1} + \cdots + \phi_p y_{t-p} + \omega_{11} x_{1(t-1)} + \cdots + \omega_{pm} x_{m(t-j)} + \varepsilon_t \tag{4}\]
Including lagged values of \(y_t\) in the model is one of the most effective remedies for serial correlation. The ADL model is the foundation of the General-to-Specific modelling strategy covered later in this chapter.
Ordinary Least Squares (OLS) is the standard method for estimating the parameters of a linear regression model. Its main objective is to minimise the sum of squared errors:
\[\text{Minimise} \quad \sum_{t=1}^{n} \varepsilon_t^2 = \sum_{t=1}^{n} (y_t - \hat{y}_t)^2\]
where \(\varepsilon_t = y_t - \hat{y}_t\) is the residual at time \(t\).
Consider a regression model with one independent variable:
\[y_t = \beta_0 + \beta_1 x_{1t} + \varepsilon_t \tag{5}\]
The fitted equation is:
\[\hat{y}_t = \hat{\beta}_0 + \hat{\beta}_1 x_{1t} \tag{6}\]
The residual is:
\[\varepsilon_t = y_t - \hat{y}_t = y_t - (\hat{\beta}_0 + \hat{\beta}_1 x_{1t}) \tag{7, 8}\]
The sum of squared errors to minimise:
\[\sum_{i=1}^{n} \varepsilon_i^2 = \sum_{i=1}^{n} \left[y_t - \hat{\beta}_0 - \hat{\beta}_1 x_{1t}\right]^2 \tag{9}\]
Partial differentiation with respect to \(\beta_0\), equating to zero:
\[\frac{\partial \sum_{t=1}^{n} \varepsilon_t^2}{\partial \beta_0} = 2\sum_{t=1}^{n}\left(y_t - \hat{\beta}_0 - \hat{\beta}_1 x_{1t}\right)(-1) = 0 \tag{10}\]
This simplifies (through equations 11–12) to the first normal equation:
\[\sum_{t=1}^{n} y_t = n\hat{\beta}_0 + \hat{\beta}_1 \sum_{t=1}^{n} x_{1t} \tag{13}\]
Partial differentiation with respect to \(\beta_1\), equating to zero:
\[\frac{\partial \sum_{t=1}^{n} \varepsilon_t^2}{\partial \beta_1} = \sum_{t=1}^{n}\left(y_t - \hat{\beta}_0 - \hat{\beta}_1 x_{1t}\right)(-x_{1t}) = 0 \tag{14}\]
This gives the second normal equation:
\[\sum_{t=1}^{n} x_{1t} y_t = \hat{\beta}_0 \sum_{t=1}^{n} x_{1t} + \hat{\beta}_1 \sum_{t=1}^{n} x_{1t}^2 \tag{16}\]
Solving these two normal equations simultaneously yields the OLS estimates \(\hat{\beta}_0\) and \(\hat{\beta}_1\).
In practice, R’s lm() function performs all of these calculations internally using matrix algebra. Understanding the derivation helps you interpret what the function is doing under the hood and why the assumptions matter.
Variables included in the model must satisfy two conditions:
Variable selection is the procedure of choosing a subset (from all possible subsets) of independent variables. The common criterion is to minimise the squared error while maintaining theoretical coherence.
| Step | Description |
|---|---|
| Step 1 | Identify the appropriate economic theory to select independent variables that explain the dependent variable. |
| Step 2 | Ensure sufficient data are available. Rule of thumb: at least 5 observations per independent variable. |
| Step 3 | Examine the relationships among variables using historical data to assess the general fitness of the model. |
A key practical challenge in multi-variable forecasting:
\[\hat{y}_{t+1} = \hat{\beta}_0 + \hat{\beta}_1 x_{1(t+1)} + \hat{\beta}_2 x_{2(t+1)}\]
To forecast \(y_{t+1}\), the analyst must also have (or forecast) \(x_{1(t+1)}\) and \(x_{2(t+1)}\).
Challenge: If the future values of independent variables are themselves estimated with error, those errors propagate into the forecast of \(y\). This is a key limitation of econometric forecasting compared to univariate methods — more variables means more sources of forecast error.
For OLS estimates to have desirable properties (unbiasedness, efficiency), the following model-level assumptions must hold:
A model is correctly specified if it includes the correct set of independent variables. Common misspecification situations are:
| Type | Description |
|---|---|
| Under-specification | Relevant variables are omitted from the model. |
| Over-specification | Irrelevant variables are included in the model. |
| Wrong functional form | Linear model used when the true relationship is non-linear. |
| Poor residuals | Diagnostic checks on residuals indicate systematic patterns. |
Two main strategies exist for estimating an econometric model:
| Strategy | Description |
|---|---|
| General-to-Specific | Start with the most general model and systematically remove insignificant variables. Also called backward stepwise. |
| Specific-to-General | Start with a simple model and add variables one at a time. Also called forward stepwise. |
The General-to-Specific approach is based on the principle of parsimony:
If there are two or more competing explanations of the same phenomenon, each having the same explanatory power, choose the simpler one.
Procedure:
\[y_t = \beta_0 + \sum_{m=1}^{k} \beta_m x_{mt} + \sum_{j=1}^{p} \phi_j y_{t-j} + \sum_{m=1}^{k}\sum_{j=1}^{q} \omega_{mj} x_{m(t-j)} + \varepsilon_t\]
Why drop one variable at a time? The significance of one variable depends on what other variables are in the model. Dropping multiple variables simultaneously can lead to incorrect conclusions about which variables truly matter.
Practical considerations for the general model:
Procedure:
Start with the simplest model, e.g.: \[y_t = \beta_0 + \beta_1 x_{1t} + \varepsilon_t \quad \text{or} \quad y_t = \phi_0 + \phi_1 y_{t-1} + \varepsilon_t\]
If the model is inadequate or mis-specified, add the next most promising variable (based on correlation with \(y_t\)).
Repeat until a well-specified model is obtained.
In this course, we primarily use the General-to-Specific approach, as illustrated in the worked example with Malaysian car registration data.
Failure to satisfy any of the following tests indicates that a model assumption is violated. Remedial action should then be taken: reformulate the model, include new variables, obtain more data, or apply variable transformations.
| Test | Purpose | Decision Rule |
|---|---|---|
| F-test | Overall model fitness | If \(p < 0.05\), at least one variable is significant |
| t-test | Significance of each coefficient | If \(p < 0.05\), retain the variable |
| Adjusted \(R^2\) | Goodness of fit | Higher is better; preferred over \(R^2\) |
| Breusch-Pagan | Heteroscedasticity | If \(p > 0.05\), variances are constant (no problem) |
| Durbin-Watson | Serial correlation | DW \(\approx\) 2 and \(p > 0.05\) means no problem |
| VIF | Multicollinearity | VIF \(> 10\) indicates a serious problem |
The F-test evaluates the overall significance of the model:
Decision: If the \(p\)-value from the F-test is less than 0.05, the model has overall fit and at least one variable is significant. Proceed to examine individual coefficients using t-tests. Otherwise, the model is considered poorly fit.
After confirming the model is overall significant via the F-test, examine each individual coefficient \(\beta_1, \beta_2, \ldots, \beta_m\):
Decision: If \(p < 0.05\), retain the variable. Otherwise, consider removing it from the model.
The coefficient of determination \(R^2\) measures the proportion of total variation in \(y_t\) explained by the regression:
\[R^2 = 1 - \frac{\text{SSR}}{\text{SST}} = 1 - \frac{\sum_{t=1}^{n}(y_t - \hat{y}_t)^2}{\sum_{t=1}^{n}(y_t - \bar{y})^2}\]
\(R^2 \in [0, 1]\), where 0 indicates no fit and 1 indicates perfect fit.
Problem with \(R^2\): It increases mechanically whenever a new variable is added, even if the variable adds no explanatory value. Therefore, use Adjusted \(R^2\):
\[\bar{R}^2 = 1 - \frac{(1 - R^2)(n - 1)}{n - m - 1}\]
where \(n\) is the number of observations and \(m\) is the number of independent variables. Adjusted \(R^2\) penalises for adding variables and is the preferred measure of goodness of fit.
Homoscedasticity means the variance of the error term is constant: \[\text{Var}(\varepsilon_t) = \sigma^2 \quad \text{for all } t\]
If this assumption is violated, the errors are heteroscedastic — their variance changes over time or across observations. Consequences:
Breusch-Pagan Test: - \(H_0\): Error variance is constant (homoscedastic). - \(H_1\): Error variance is non-constant (heteroscedastic). - Decision: If \(p > 0.05\), no heteroscedasticity problem. If \(p < 0.05\), heteroscedasticity is present — consider transforming the dependent variable (e.g., log transformation).
The error terms must be uncorrelated across time: \[\text{Cov}(\varepsilon_t, \varepsilon_{t-s}) = 0 \quad \text{for all } s \neq 0\]
The most common form is first-order autocorrelation: \[\varepsilon_t = \rho \varepsilon_{t-1} + \nu_t \tag{17}\]
By repeated substitution of equation (17):
\[\varepsilon_t = \rho \varepsilon_{t-1} + \nu_t = \rho^k \varepsilon_{t-k} + \sum_{j=0}^{k-1} \rho^j \nu_{t-j} \tag{18}\]
Under perfect correlation (\(\rho = 1\)): \[\varepsilon_t = \varepsilon_{t-k} + \sum_{j=1}^{k-1} \nu_{t-j} \tag{19}\]
\[\text{Var}(\varepsilon_t) = \sigma^2_\varepsilon + k\sigma^2_\nu\]
Since the variance of \(\varepsilon_t\) grows with \(k\), forecast accuracy deteriorates as the forecast horizon increases. When \(\rho = 0\), \(\text{Var}(\varepsilon_t) = \sigma^2_\varepsilon\) is constant and there is no serial correlation.
\[DW = \frac{\sum_{t=2}^{n}(\varepsilon_t - \varepsilon_{t-1})^2}{\sum_{t=1}^{n} \varepsilon_t^2} \tag{20}\]
| DW Value | Interpretation |
|---|---|
| DW \(< 2\) | Positive serial correlation |
| DW \(\approx 2\) | No serial correlation |
| DW \(> 2\) | Negative serial correlation |
Multicollinearity occurs when two or more independent variables are linearly related to each other.
Consider \(y_t = \beta_0 + \beta_1 x_{1t} + \beta_2 x_{2t} + \varepsilon_t\) where \(x_{1t} = 2x_{2t}\):
\[y_t = \beta_0 + 2\beta_1 x_{2t} + \beta_2 x_{2t} + \varepsilon_t = \beta_0 + (2\beta_1 + \beta_2)x_{2t} + \varepsilon_t\]
It is impossible to separate the individual effects of \(x_{1t}\) and \(x_{2t}\) — their contributions are confounded.
\[\text{VIF}_i = \frac{1}{1 - R_i^2}\]
where \(R_i^2\) is the \(R^2\) from regressing variable \(i\) on all other independent variables.
| VIF | Interpretation |
|---|---|
| VIF \(< 5\) | No problem |
| \(5 \leq\) VIF \(< 10\) | Moderate concern |
| VIF \(\geq 10\) | Serious multicollinearity — action required |
This section demonstrates the complete step-by-step procedure for building an econometric model for forecasting purposes.
Objective: Develop an econometric model to forecast the demand for cars in Malaysia, measured by the Number of New Car Registrations (’000).
knitr::kable(head(econ), digits = 2)| Year | Number of Car Registration (’000) | BLR (%) | Unemployment Rate (%) | Total Population | CPI | Total Export | Income | GDP |
|---|---|---|---|---|---|---|---|---|
| 1969 | 21.60 | 9.1 | 7.6 | 10061684 | -0.41 | 1.73 | 264 | 17817535563 |
| 1970 | 23.10 | 9.2 | 7.5 | 10306508 | 1.84 | 1.76 | 264 | 18884189212 |
| 1971 | 25.80 | 9.2 | 7.5 | 10552557 | 1.61 | 1.72 | 264 | 20779153503 |
| 1972 | 29.40 | 8.9 | 7.4 | 10801619 | 3.23 | 1.82 | 264 | 22729992899 |
| 1973 | 31.20 | 9.9 | 7.2 | 11062664 | 10.56 | 3.18 | 264 | 25389647919 |
| 1974 | 41.61 | 11.2 | 7.2 | 11335187 | 17.33 | 4.59 | 362 | 27501726906 |
ggplot(econ, aes(x = Year, y = `Number of Car Registration ('000)`)) +
geom_line(colour = "steelblue", linewidth = 0.9) +
geom_point(colour = "steelblue", size = 1.8) +
labs(title = "New Car Registrations in Malaysia",
x = "Year",
y = "Number ('000)") +
theme_ts()
# Check the exact column names before modelling
names(econ)[1] "Year" "Number of Car Registration ('000)"
[3] "BLR (%)" "Unemployment Rate (%)"
[5] "Total Population" "CPI"
[7] "Total Export" "Income"
[9] "GDP" The dataset contains 54 annual observations from 1969 to 2022. Based on economic theory, the expected relationships between each independent variable and new car registrations are:
| Independent Variable | Expected Sign | Rationale |
|---|---|---|
| Unemployment Rate (%) | Negative | Higher unemployment reduces household income and car purchases |
| CPI | Negative | Higher prices reduce real purchasing power |
| BLR (%) | Negative | Higher lending rates raise financing costs, dampening demand |
| GDP | Positive | Higher national income supports consumer spending |
| Total Export | Positive | Export growth reflects economic prosperity |
| Total Population | Positive | More people, more demand for cars |
| Income | Positive | Higher per capita income directly boosts car demand |
# Create lag 1 of number of car registrations
# This is needed to address serial correlation (see Model 2 onwards)
lag1 <- c(NA, econ$`Number of Car Registration ('000)`[
1:(length(econ$`Number of Car Registration ('000)`) - 1)])The first model includes all seven independent variables. This is the general model in the General-to-Specific procedure.
\[\hat{y}_t = \hat{\beta}_0 + \hat{\beta}_1(\text{Unemployment}) + \hat{\beta}_2(\text{CPI}) + \hat{\beta}_3(\text{BLR}) + \hat{\beta}_4(\text{GDP}) + \hat{\beta}_5(\text{Export}) + \hat{\beta}_6(\text{Population}) + \hat{\beta}_7(\text{Income})\]
# Model 1: full model with all independent variables
# Dependent variable Y: Number of Car Registration ('000)
# Independent variables X: all economic predictors
model1 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + CPI + `BLR (%)` +
GDP + `Total Export` + `Total Population` + Income,
data = econ)
summary(model1)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
CPI + `BLR (%)` + GDP + `Total Export` + `Total Population` +
Income, data = econ)
Residuals:
Min 1Q Median 3Q Max
-133.033 -25.599 6.262 18.042 103.360
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.108e+02 1.343e+02 0.825 0.41347
`Unemployment Rate (%)` -2.070e+01 7.653e+00 -2.705 0.00953 **
CPI 4.406e-01 2.598e+00 0.170 0.86604
`BLR (%)` -8.711e+00 6.251e+00 -1.394 0.17016
GDP -1.601e-09 1.156e-09 -1.385 0.17277
`Total Export` 1.768e+00 3.860e-01 4.579 3.55e-05 ***
`Total Population` 1.454e-05 6.130e-06 2.372 0.02192 *
Income 3.715e-02 3.401e-02 1.092 0.28039
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 46.27 on 46 degrees of freedom
Multiple R-squared: 0.9673, Adjusted R-squared: 0.9623
F-statistic: 194.5 on 7 and 46 DF, p-value: < 2.2e-16# Diagnostic tests for Model 1
cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model1)`Unemployment Rate (%)` CPI `BLR (%)`
3.849078 1.393722 7.000983
GDP `Total Export` `Total Population`
414.470181 34.029012 54.943368
Income
170.152327 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model1) lag Autocorrelation D-W Statistic p-value
1 0.4470231 1.101488 0
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model1)
studentized Breusch-Pagan test
data: model1
BP = 7.8119, df = 7, p-value = 0.3495p1 <- ggplot(data.frame(Fitted = fitted(model1), Residuals = residuals(model1)),
aes(x = Fitted, y = Residuals)) +
geom_point(colour = "steelblue", alpha = 0.7) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
labs(title = "Residuals vs Fitted", x = "Fitted values", y = "Residuals") +
theme_ts()
p2 <- ggplot(data.frame(Residuals = residuals(model1)), aes(sample = Residuals)) +
stat_qq(colour = "steelblue") +
stat_qq_line(colour = "red", linetype = "dashed") +
labs(title = "Normal Q-Q Plot") +
theme_ts()
grid.arrange(p1, p2, ncol = 2)
Model 1 Interpretation:
Action: Add the lag variable of new car registration to address the serial correlation.
Adding \(y_{t-1}\) (lag 1 of car registrations) is the standard remedy for serial correlation.
\[\hat{y}_t = \hat{\beta}_0 + \hat{\beta}_1(\text{Unemployment}) + \hat{\beta}_2(\text{CPI}) + \hat{\beta}_3(\text{BLR}) + \hat{\beta}_4(\text{GDP}) + \hat{\beta}_5(\text{Export}) + \hat{\beta}_6(\text{Population}) + \hat{\beta}_7(\text{Income}) + \hat{\phi}_1 y_{t-1}\]
# Model 2: add lag1 to address serial correlation
model2 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + CPI + `BLR (%)` +
GDP + `Total Export` + `Total Population` + Income + lag1,
data = econ)
summary(model2)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
CPI + `BLR (%)` + GDP + `Total Export` + `Total Population` +
Income + lag1, data = econ)
Residuals:
Min 1Q Median 3Q Max
-83.653 -14.255 0.009 13.669 94.445
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.411e+02 1.094e+02 1.290 0.203828
`Unemployment Rate (%)` -1.473e+01 6.132e+00 -2.402 0.020587 *
CPI 3.344e-01 2.154e+00 0.155 0.877331
`BLR (%)` -4.880e+00 4.990e+00 -0.978 0.333483
GDP -9.560e-10 9.200e-10 -1.039 0.304454
`Total Export` 1.196e+00 3.232e-01 3.702 0.000593 ***
`Total Population` 3.357e-06 5.464e-06 0.614 0.542141
Income 2.357e-02 2.693e-02 0.875 0.386057
lag1 5.119e-01 9.389e-02 5.453 2.14e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 36.48 on 44 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.98, Adjusted R-squared: 0.9764
F-statistic: 269.6 on 8 and 44 DF, p-value: < 2.2e-16cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model2)`Unemployment Rate (%)` CPI `BLR (%)`
3.694196 1.493748 7.158650
GDP `Total Export` `Total Population`
411.683506 37.556426 67.445528
Income lag1
168.195560 18.671873 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model2) lag Autocorrelation D-W Statistic p-value
1 -0.002775646 1.853213 0.152
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model2)
studentized Breusch-Pagan test
data: model2
BP = 18.234, df = 8, p-value = 0.01954Model 2 Interpretation:
Action: Drop GDP (most insignificant with highest VIF) from the model.
# Model 3: remove GDP (most insignificant + highest VIF)
model3 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + CPI + `BLR (%)` +
`Total Export` + `Total Population` + Income + lag1,
data = econ)
summary(model3)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
CPI + `BLR (%)` + `Total Export` + `Total Population` + Income +
lag1, data = econ)
Residuals:
Min 1Q Median 3Q Max
-77.268 -16.073 -0.469 11.554 104.394
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.365e+02 1.094e+02 1.248 0.218566
`Unemployment Rate (%)` -1.304e+01 5.918e+00 -2.204 0.032701 *
CPI 1.965e-02 2.134e+00 0.009 0.992694
`BLR (%)` -3.492e+00 4.812e+00 -0.726 0.471846
`Total Export` 9.986e-01 2.614e-01 3.820 0.000406 ***
`Total Population` 4.600e-07 4.703e-06 0.098 0.922522
Income -3.264e-03 7.613e-03 -0.429 0.670121
lag1 5.248e-01 9.316e-02 5.633 1.09e-06 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 36.51 on 45 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9795, Adjusted R-squared: 0.9763
F-statistic: 307.4 on 7 and 45 DF, p-value: < 2.2e-16cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model3)`Unemployment Rate (%)` CPI `BLR (%)`
3.434997 1.464204 6.645678
`Total Export` `Total Population` Income
24.520363 49.886310 13.422505
lag1
18.349569 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model3) lag Autocorrelation D-W Statistic p-value
1 0.03327008 1.751804 0.068
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model3)
studentized Breusch-Pagan test
data: model3
BP = 14.423, df = 7, p-value = 0.04416Model 3 Interpretation:
Action: Drop Income from the model.
# Model 4: remove Income (incorrect sign + most insignificant)
model4 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + `BLR (%)` +
`Total Export` + `Total Population` + CPI + lag1,
data = econ)
summary(model4)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
`BLR (%)` + `Total Export` + `Total Population` + CPI + lag1,
data = econ)
Residuals:
Min 1Q Median 3Q Max
-79.452 -18.570 -0.189 12.885 99.914
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.610e+02 9.250e+01 1.740 0.088530 .
`Unemployment Rate (%)` -1.420e+01 5.219e+00 -2.721 0.009169 **
`BLR (%)` -3.609e+00 4.762e+00 -0.758 0.452389
`Total Export` 9.939e-01 2.588e-01 3.840 0.000374 ***
`Total Population` -8.518e-07 3.540e-06 -0.241 0.810923
CPI -1.456e-01 2.080e+00 -0.070 0.944521
lag1 5.297e-01 9.163e-02 5.781 6.17e-07 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 36.19 on 46 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9794, Adjusted R-squared: 0.9768
F-statistic: 365.1 on 6 and 46 DF, p-value: < 2.2e-16cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model4)`Unemployment Rate (%)` `BLR (%)` `Total Export`
2.720139 6.624291 24.477286
`Total Population` CPI lag1
28.777339 1.416478 18.072527 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model4) lag Autocorrelation D-W Statistic p-value
1 0.04457492 1.745045 0.086
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model4)
studentized Breusch-Pagan test
data: model4
BP = 13.282, df = 6, p-value = 0.03877Model 4 Interpretation:
Action: Remove Total Population from the model.
# Model 5: remove Total Population (incorrect sign + insignificant)
model5 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + `BLR (%)` +
`Total Export` + CPI + lag1,
data = econ)
summary(model5)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
`BLR (%)` + `Total Export` + CPI + lag1, data = econ)
Residuals:
Min 1Q Median 3Q Max
-81.417 -17.893 0.529 12.971 98.928
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 146.07589 68.07994 2.146 0.0371 *
`Unemployment Rate (%)` -13.82127 4.92741 -2.805 0.0073 **
`BLR (%)` -3.49417 4.69007 -0.745 0.4600
`Total Export` 0.95831 0.21034 4.556 3.71e-05 ***
CPI 0.03282 1.92420 0.017 0.9865
lag1 0.52062 0.08273 6.293 9.67e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 35.82 on 47 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9794, Adjusted R-squared: 0.9772
F-statistic: 447.1 on 5 and 47 DF, p-value: < 2.2e-16cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model5)`Unemployment Rate (%)` `BLR (%)` `Total Export`
2.474048 6.558008 16.497116
CPI lag1
1.236596 15.033042 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model5) lag Autocorrelation D-W Statistic p-value
1 0.05141295 1.734915 0.124
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model5)
studentized Breusch-Pagan test
data: model5
BP = 12.122, df = 5, p-value = 0.03315Model 5 Interpretation:
Action: Remove CPI from the model.
# Model 6: remove CPI (most insignificant)
model6 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + `BLR (%)` +
`Total Export` + lag1,
data = econ)
summary(model6)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
`BLR (%)` + `Total Export` + lag1, data = econ)
Residuals:
Min 1Q Median 3Q Max
-81.403 -17.873 0.738 13.383 98.933
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 146.08660 67.36439 2.169 0.03510 *
`Unemployment Rate (%)` -13.82329 4.87442 -2.836 0.00667 **
`BLR (%)` -3.47933 4.56045 -0.763 0.44924
`Total Export` 0.95889 0.20540 4.669 2.47e-05 ***
lag1 0.52038 0.08062 6.455 5.04e-08 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 35.45 on 48 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9794, Adjusted R-squared: 0.9777
F-statistic: 570.8 on 4 and 48 DF, p-value: < 2.2e-16cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model6)`Unemployment Rate (%)` `BLR (%)` `Total Export`
2.472613 6.332415 16.065072
lag1
14.580335 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model6) lag Autocorrelation D-W Statistic p-value
1 0.05149487 1.734734 0.15
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model6)
studentized Breusch-Pagan test
data: model6
BP = 11.858, df = 4, p-value = 0.01844Model 6 Interpretation:
Action: Remove Total Export from the model.
# Model 7: final model — remove Total Export
# Remaining variables: Unemployment Rate, BLR, lag1
model7 <- lm(`Number of Car Registration ('000)` ~
`Unemployment Rate (%)` + `BLR (%)` + lag1,
data = econ)
summary(model7)
Call:
lm(formula = `Number of Car Registration ('000)` ~ `Unemployment Rate (%)` +
`BLR (%)` + lag1, data = econ)
Residuals:
Min 1Q Median 3Q Max
-86.523 -20.412 -0.523 17.512 180.560
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 264.93628 74.43585 3.559 0.000838 ***
`Unemployment Rate (%)` -16.68328 5.77138 -2.891 0.005715 **
`BLR (%)` -13.84240 4.75453 -2.911 0.005402 **
lag1 0.78071 0.06949 11.235 3.66e-15 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 42.31 on 49 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared: 0.9701, Adjusted R-squared: 0.9682
F-statistic: 529.2 on 3 and 49 DF, p-value: < 2.2e-16cat("--- VIF (Multicollinearity) ---\n")--- VIF (Multicollinearity) ---vif(model7)`Unemployment Rate (%)` `BLR (%)` lag1
2.433558 4.832143 7.604651 cat("\n--- Durbin-Watson Test (Serial Correlation) ---\n")
--- Durbin-Watson Test (Serial Correlation) ---durbinWatsonTest(model7) lag Autocorrelation D-W Statistic p-value
1 -0.06392779 1.755684 0.192
Alternative hypothesis: rho != 0cat("\n--- Breusch-Pagan Test (Heteroscedasticity) ---\n")
--- Breusch-Pagan Test (Heteroscedasticity) ---bptest(model7)
studentized Breusch-Pagan test
data: model7
BP = 4.4204, df = 3, p-value = 0.2195p3 <- ggplot(data.frame(Fitted = fitted(model7), Residuals = residuals(model7)),
aes(x = Fitted, y = Residuals)) +
geom_point(colour = "steelblue", alpha = 0.7) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
labs(title = "Residuals vs Fitted (Model 7)", x = "Fitted values", y = "Residuals") +
theme_ts()
p4 <- ggplot(data.frame(Residuals = residuals(model7)), aes(sample = Residuals)) +
stat_qq(colour = "steelblue") +
stat_qq_line(colour = "red", linetype = "dashed") +
labs(title = "Normal Q-Q Plot (Model 7)") +
theme_ts()
grid.arrange(p3, p4, ncol = 2)
fit_df <- data.frame(
Year = econ$Year[!is.na(lag1)],
Actual = econ$`Number of Car Registration ('000)`[!is.na(lag1)],
Fitted = fitted(model7)
)
ggplot(fit_df, aes(x = Year)) +
geom_line(aes(y = Actual, colour = "Actual"), linewidth = 0.9) +
geom_line(aes(y = Fitted, colour = "Fitted"),
linewidth = 0.9, linetype = "dashed") +
scale_colour_manual(values = c("Actual" = "steelblue", "Fitted" = "#d73027")) +
labs(title = "New Car Registrations: Actual vs Fitted (Model 7)",
x = "Year", y = "Number ('000)", colour = NULL) +
theme_ts()
Model 7 Interpretation (Final Model):
The final estimated model is:
\[\hat{y}_t = 264.94 - 16.68\, x_{1t} - 13.84\, x_{2t} + 0.78\, y_{t-1}\]
where:
| Coefficient | Value | Interpretation |
|---|---|---|
| Intercept | 264.94 | Baseline car registrations when all predictors are zero |
| Unemployment Rate | \(-16.68\) | A 1 percentage-point increase in unemployment reduces car demand by approximately 16,680 units |
| BLR | \(-13.84\) | A 1 percentage-point increase in BLR reduces car demand by approximately 13,840 units |
| Lag of car registrations | \(+0.78\) | If car registrations increased by 1,000 last year, demand this year increases by approximately 780 units |
All signs are consistent with economic theory: higher unemployment and higher borrowing costs both suppress consumer demand for cars, while past registration levels exhibit strong positive persistence.
Given:
# Extract coefficients from the final model
coefs <- coef(model7)
cat("Model coefficients:\n")Model coefficients:print(round(coefs, 4)) (Intercept) `Unemployment Rate (%)` `BLR (%)`
264.9363 -16.6833 -13.8424
lag1
0.7807 # Future values of independent variables for 2023
unemp_2023 <- 3.3 # Unemployment Rate (%)
blr_2023 <- 6.89 # BLR (%)
y_2022 <- 744.78 # Car registrations in 2022 ('000)
# One-step-ahead forecast
y_hat_2023 <- coefs["(Intercept)"] +
coefs["`Unemployment Rate (%)`"] * unemp_2023 +
coefs["`BLR (%)`"] * blr_2023 +
coefs["lag1"] * y_2022
cat("\nForecast for 2023 (number of new car registrations, '000):",
round(y_hat_2023, 2), "\n")
Forecast for 2023 (number of new car registrations, '000): 695.96 cat("Forecast in actual units:", round(y_hat_2023 * 1000), "\n")Forecast in actual units: 695961 Forecasting caveat: The quality of this forecast depends on how accurately we know (or can forecast) the unemployment rate and BLR for 2023. If those values are themselves uncertain, the forecast error will be larger than the model’s in-sample statistics suggest.
model_summary <- data.frame(
Model = paste0("Model ", 1:7),
Variables_Dropped = c(
"None (full model)",
"— (add lag1)",
"GDP removed",
"Income removed",
"Total Population removed",
"CPI removed",
"Total Export removed"
),
Adj_R2 = c(
round(summary(model1)$adj.r.squared, 4),
round(summary(model2)$adj.r.squared, 4),
round(summary(model3)$adj.r.squared, 4),
round(summary(model4)$adj.r.squared, 4),
round(summary(model5)$adj.r.squared, 4),
round(summary(model6)$adj.r.squared, 4),
round(summary(model7)$adj.r.squared, 4)
),
DW_pval = c(
round(durbinWatsonTest(model1)$p, 3),
round(durbinWatsonTest(model2)$p, 3),
round(durbinWatsonTest(model3)$p, 3),
round(durbinWatsonTest(model4)$p, 3),
round(durbinWatsonTest(model5)$p, 3),
round(durbinWatsonTest(model6)$p, 3),
round(durbinWatsonTest(model7)$p, 3)
),
BP_pval = c(
round(bptest(model1)$p.value, 3),
round(bptest(model2)$p.value, 3),
round(bptest(model3)$p.value, 3),
round(bptest(model4)$p.value, 3),
round(bptest(model5)$p.value, 3),
round(bptest(model6)$p.value, 3),
round(bptest(model7)$p.value, 3)
)
)
knitr::kable(model_summary,
col.names = c("Model", "Change from Previous",
"Adj. R²", "DW p-value", "BP p-value"),
align = "lllll")| Model | Change from Previous | Adj. R<U+00B2> | DW p-value | BP p-value |
|---|---|---|---|---|
| Model 1 | None (full model) | 0.9623 | 0.000 | 0.349 |
| Model 2 | <U+2014> (add lag1) | 0.9764 | 0.156 | 0.020 |
| Model 3 | GDP removed | 0.9763 | 0.082 | 0.044 |
| Model 4 | Income removed | 0.9768 | 0.110 | 0.039 |
| Model 5 | Total Population removed | 0.9772 | 0.122 | 0.033 |
| Model 6 | CPI removed | 0.9777 | 0.176 | 0.018 |
| Model 7 | Total Export removed | 0.9682 | 0.194 | 0.220 |
This chapter covered the complete framework for econometric modelling in time series forecasting. The key takeaways are:
Model Structure: Econometric models relate a dependent variable to multiple independent (explanatory) variables, grounded in economic theory. The ADL model incorporates both current and lagged variables.
OLS Estimation: Ordinary Least Squares minimises the sum of squared residuals. Understanding its derivation helps interpret model output and diagnose problems.
Issues in Model Construction: Variable selection must balance statistical significance with theoretical justification. Practical challenges include model cost, complexity, and the need for future values of predictors.
Assumptions: Five key assumptions govern OLS — linearity, correct specification, no multicollinearity, homoscedasticity, and no serial correlation. Violations require diagnostic tests and remedial action.
Model Estimation Strategies:
Statistical Validation: A valid model must pass six diagnostic tests: F-test, t-tests, adjusted \(R^2\), Breusch-Pagan (heteroscedasticity), Durbin-Watson (serial correlation), and VIF (multicollinearity).
Worked Example: The final model for Malaysian new car registrations identified Unemployment Rate, BLR, and the lag of car registrations as the significant determinants. The model satisfies all diagnostic conditions and produces an interpretable, parsimonious equation.