Econometría II

name: xaringan-title
class: inverse, left, bottom
background-image: url(pictures/picuniform.jpg)
background-size: cover

# **Econometría II**
----

## **<br/> ARMAX**

### Carlos A. Yanes Guerra
### 2025

---
class: inverse, middle, center

# Preguntas de las sesiones anteriores?

---
layout: true
# Modelos ARMAX

---

- **ARMAX** (AutoRegressive Moving Average with eXogenous inputs) es un modelo de series de tiempo que combina tres componentes:
  - **AR**: Componente autorregresivo.
  - **MA**: Componente de medias móviles.
  - **X**: Entrada exógena, es decir, variables externas que influyen en el sistema.

La .hi[ecuación general] del modelo ARMAX es:

`$$y_t = \phi_1 y_{t-1} + \phi_2 y_{t-2} + ... + \phi_p y_{t-p} + \theta_1 e_{t-1} + \theta_2 e_{t-2} + ... + \theta_q e_{t-q} + \beta_1 x_{t-1} + \beta_2 x_{t-2} + ... + \beta_r x_{t-r} + e_t$$`

---

1. **Identificación**: Examinar los gráficos de autocorrelación (ACF) y autocorrelación parcial (PACF) para determinar los posibles órdenes $ p $ y $ q $ de los componentes AR y MA.
2. **Selección de Variables Exógenas**: Identificar las variables exógenas relevantes $ x_t $.
3. **Ajuste del Modelo**: Estimar los parámetros del modelo mediante métodos como MLE (Maximum Likelihood Estimation) o mínimos cuadrados.
4. **Diagnóstico**: Verificar los residuos del modelo para asegurarse de que no hay patrones no modelados.

---

- **Estacionariedad**: Asegúrate de que las series de tiempo sean estacionarias antes de aplicar el modelo ARMAX.
- **Relación entre Variables Exógenas**: Las variables exógenas deben tener una relación significativa con la variable dependiente.
- **Modelos de Residuos**: Los residuos del modelo deben ser ruido blanco (sin autocorrelación).

### Aplicaciones de ARMAX

- **Previsión Económica**: En econometría y macroeconomía, ARMAX se utiliza para predecir series de tiempo como el PIB, inflación, etc., incorporando factores externos como políticas monetarias.
- **Control de Procesos**: En ingeniería y sistemas de control, ARMAX es útil para modelar sistemas dinámicos donde las entradas externas afectan el comportamiento del sistema.
- **Modelos Financieros**: Análisis de activos financieros, donde las variables externas (como tasas de interés) influyen en los precios de las acciones.

---

---

<cy-blockquote>Teniendo como referencia los modelos univariados, podemos entonces involucrar de manera exogena, la participación de un componente **exogeno** que acompaña nuestro modelo</cy-blockquote>

`$$y_t=\color{red}{c}+\color{purple}{v(\beta)x_t}+ e_t$$`

Dicho de otra manera:

`$$y_t=C_0+\sum \limits_{i=1}^p \phi_i y_{t-1}+\color{purple}{\sum \limits_{k=1}^r \beta_k x_{t}}+\color{blue}{\sum \limits_{j=1}^q \theta_i e_{t-1}}+ \color{red}{e_t}$$`

*La parte de x, es exogena e incluso se vincula como variable en función de (t), es decir en el presente*.

---

Los modelos de tipo .hi[ARMAX] son usados en agricultura, energia, por ejemplo el siguiente paper de [Rabbi, et.al (2020)](https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9350326&tag=1) nos muestra algo de ello.

Un modelo ARMAX suele ser representado como:

`$$Y_{t}=\alpha_{0} +\alpha_{1}Y_{t-1}+\cdots+\alpha_{\rho}Y_{t-\rho}+\beta_{1}X_{t}+\epsilon_{t}$$`

Donde:

`$X_{t}$` es la variable exogena o explicativa y el error del modelo debe tener la estimación:

`$$\epsilon_{t}= \rho \epsilon_{t-1}+ \theta_{1}V_{t-1}+ V_{t}$$`
Que vendría a ser un proceso .hi[ARMA] para el .hi-red[residuo] del modelo original.

---

Un modelo autoregresivo (AR) y de media móvil (MA) puede contener variables exógenas, siempre y cuando el proceso `$(Y_{t})$` sea una solución .hi-purple[estacionaria] de ecuaciones en diferencia de tal forma que:

`$$Y_{t} - \alpha_{1} Y_{t-1}- \cdots-\alpha_{\rho} Y_{t-\rho} = Z_{t}+ \Theta_{1} Z_{t-1}+\cdots + \Theta_{q} Z_{t-q}$$`

Donde tanto `$\alpha$` y `$\Theta$` son parámetros de las variables y `$Z_{t}$` es una variable exógena que se distribuye R.B `$\sim (0, \sigma^{2})$`

---
layout: false
.attn[Ejemplo:]

---

---
layout: true
# Ejemplo

---

### Condiciones de estimación

---

.pull-left[

```
#> 
#> ####################### 
#> # KPSS Unit Root Test # 
#> ####################### 
#> 
#> Test is of type: tau with 4 lags. 
#> 
#> Value of test-statistic is: 0.0887 
#> 
#> Critical value for a significance level of: 
#>                 10pct  5pct 2.5pct  1pct
#> critical values 0.119 0.146  0.176 0.216
```
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:red;overflow:visible;position:relative;"><path d="M334.5 414c8.8 3.8 19 2 26-4.6l144-136c4.8-4.5 7.5-10.8 7.5-17.4s-2.7-12.9-7.5-17.4l-144-136c-7-6.6-17.2-8.4-26-4.6s-14.5 12.5-14.5 22l0 72L32 192c-17.7 0-32 14.3-32 32l0 64c0 17.7 14.3 32 32 32l288 0 0 72c0 9.6 5.7 18.2 14.5 22z"/></svg> Serie .hi-purple[Consumo]
]

.pull-right[

```
#> 
#> ####################### 
#> # KPSS Unit Root Test # 
#> ####################### 
#> 
#> Test is of type: tau with 4 lags. 
#> 
#> Value of test-statistic is: 0.0601 
#> 
#> Critical value for a significance level of: 
#>                 10pct  5pct 2.5pct  1pct
#> critical values 0.119 0.146  0.176 0.216
```
<svg aria-hidden="true" role="img" viewBox="0 0 512 512" style="height:1em;width:1em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:red;overflow:visible;position:relative;"><path d="M334.5 414c8.8 3.8 19 2 26-4.6l144-136c4.8-4.5 7.5-10.8 7.5-17.4s-2.7-12.9-7.5-17.4l-144-136c-7-6.6-17.2-8.4-26-4.6s-14.5 12.5-14.5 22l0 72L32 192c-17.7 0-32 14.3-32 32l0 64c0 17.7 14.3 32 32 32l288 0 0 72c0 9.6 5.7 18.2 14.5 22z"/></svg> Serie .hi[Ingreso]
]

---

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---

``` r
ax <- Arima(uschange[,1], c(1, 0, 2), 
                xreg = uschange[,2])
summary(ax)
```

```
#> Series: uschange[, 1] 
#> Regression with ARIMA(1,0,2) errors 
#> 
#> Coefficients:
#>          ar1      ma1     ma2  intercept    xreg
#>       0.6922  -0.5758  0.1984     0.5990  0.2028
#> s.e.  0.1159   0.1301  0.0756     0.0884  0.0461
#> 
#> sigma^2 = 0.3219:  log likelihood = -156.95
#> AIC=325.91   AICc=326.37   BIC=345.29
#> 
#> Training set error measures:
#>                       ME      RMSE       MAE     MPE     MAPE      MASE
#> Training set 0.001714366 0.5597088 0.4209056 27.4477 161.8417 0.6594731
#>                     ACF1
#> Training set 0.006299231
```

.attn[Qué pasa si lo quiero con mas] (X's)

---

Toca algo como:

```r
conjunto <- ts.intersect(uschange[,2],
                         uschange[,3],
                         uschange[,4])
```

Luego solo en la parte donde va `xreg= conjunto`. Recuerde que todas las variables deben ser estacionarias. A modo de ejemplo vamos a mirar todas las variables, pero no van a ser correctamente especificado.

Por otro lado se comporta como un modelo de .hi-purple[regresión] y haremos uso del paquete `library(lmtest)`.

---

``` r
axc <- Arima(uschange[,1], c(1, 0, 2), xreg = conjunto)
summary(axc)
```

```
#> Series: uschange[, 1] 
#> Regression with ARIMA(1,0,2) errors 
#> 
#> Coefficients:
#>           ar1     ma1     ma2  intercept  uschange[, 2]  uschange[, 3]
#>       -0.3591  0.2691  0.1051     0.2361         0.7313         0.0823
#> s.e.   0.3552  0.3459  0.1013     0.0348         0.0427         0.0171
#>       uschange[, 4]
#>             -0.0460
#> s.e.         0.0028
#> 
#> sigma^2 = 0.1084:  log likelihood = -54.04
#> AIC=124.08   AICc=124.88   BIC=149.92
#> 
#> Training set error measures:
#>                        ME     RMSE       MAE      MPE    MAPE      MASE
#> Training set 0.0001104599 0.323003 0.2343151 14.62825 82.0396 0.3671239
#>                      ACF1
#> Training set -0.003148503
```

---

.pull-left[

``` r
library(lmtest)
coeftest(ax)
```

```
#> 
#> z test of coefficients:
#> 
#>            Estimate Std. Error z value  Pr(>|z|)    
#> ar1        0.692229   0.115905  5.9724 2.338e-09 ***
#> ma1       -0.575809   0.130063 -4.4272 9.549e-06 ***
#> ma2        0.198361   0.075585  2.6243  0.008682 ** 
#> intercept  0.599040   0.088398  6.7767 1.230e-11 ***
#> xreg       0.202819   0.046075  4.4019 1.073e-05 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]

.pull-right[

``` r
coeftest(axc)
```

```
#> 
#> z test of coefficients:
#> 
#>                 Estimate Std. Error  z value  Pr(>|z|)    
#> ar1           -0.3591467  0.3551614  -1.0112    0.3119    
#> ma1            0.2691179  0.3459332   0.7779    0.4366    
#> ma2            0.1050597  0.1012945   1.0372    0.2997    
#> intercept      0.2360594  0.0348330   6.7769 1.228e-11 ***
#> uschange[, 2]  0.7312921  0.0427249  17.1163 < 2.2e-16 ***
#> uschange[, 3]  0.0822889  0.0170837   4.8168 1.459e-06 ***
#> uschange[, 4] -0.0459726  0.0028476 -16.1441 < 2.2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```
]

---

```
#> 
#> 	Ljung-Box test
#> 
#> data:  Residuals from Regression with ARIMA(1,0,2) errors
#> Q* = 5.8916, df = 5, p-value = 0.3169
#> 
#> Model df: 3.   Total lags used: 8
```

---

```
#> 
#> 	Ljung-Box test
#> 
#> data:  Residuals from Regression with ARIMA(1,0,2) errors
#> Q* = 5.8916, df = 5, p-value = 0.3169
#> 
#> Model df: 3.   Total lags used: 8
```

---

.attn[Pronostico Modelo Real]

``` r
fcast <- forecast(ax,
xreg=rep(mean(uschange[,2]),8), h=8)
fcast
```

```
#>         Point Forecast        Lo 80    Hi 80      Lo 95    Hi 95
#> 2016 Q4      0.7844455  0.057363562 1.511527 -0.3275304 1.896421
#> 2017 Q1      0.7860095  0.054016921 1.518002 -0.3334766 1.905496
#> 2017 Q2      0.7732613  0.013689680 1.532833 -0.3884033 1.934926
#> 2017 Q3      0.7644367 -0.008001435 1.536875 -0.4169055 1.945779
#> 2017 Q4      0.7583280 -0.020200117 1.536856 -0.4323280 1.948984
#> 2018 Q1      0.7540994 -0.027330110 1.535529 -0.4409939 1.949193
#> 2018 Q2      0.7511722 -0.031643749 1.533988 -0.4460415 1.948386
#> 2018 Q3      0.7491459 -0.034333517 1.532625 -0.4490825 1.947374
```

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layout: false
class: inverse, middle, center

# Gracias por su atención

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# Bibliografía

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name: adios
class: middle

.pull-left[
# **¡Gracias!**
<br/>
## Modelos ARMAX

### Seguimos aprendiendo
]

.pull-right[
.right[
<img style="border-radius: 50%;"
src="https://avatars.githubusercontent.com/u/39503983?v=4"
width="150px" />