Ejercicios de modelos de Volatilidad

Indice de Producción Industrial

Apartado 1

Comprobad si hay efectos ARCH (gráficos y contrastes LM).

Si ninguna de las variables presenta efectos ARCH usad datos de precios de algún activo financiero o índice bursátil.

mx.ret <- CalculateReturns(mx, method="log")
mx.ret <- mx.ret[-1,]
dataToPlot <- cbind(mx.ret, mx.ret^2, abs(mx.ret))
colnames(dataToPlot) <- c("Returns", "Returns^2", "abs(Returns)")
plot.zoo(dataToPlot, main="IPI Quarterly Returns", col="blue")
par(mfrow=c(3,1))

acf(mx.ret, main="IPI Returns")
acf(mx.ret^2, main="IPI Returns^2")

acf(abs(mx.ret), main="IPI abs(Returns)")
par(mfrow=c(1,1)) 

table.Stats(mx.ret)

Box.test(coredata(mx.ret^2), type="Ljung-Box", lag = 12)

    Box-Ljung test

data:  coredata(mx.ret^2)
X-squared = 35.133, df = 12, p-value = 0.0004461

ArchTest(mx.ret)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  mx.ret
Chi-squared = 35.558, df = 12, p-value = 0.0003815

En los gráficos de retornos se ven claramente zonas con alta volatilidad sobretodo por 1995, en cambio en los últimos años los gráficos de retornos son más estables.

Se rechaza la hipótesis nula del LM-test y por tanto nuestra serie tiene efectos ARCH.

Apartado 2

Construid modelos de la familia GARCH para las variables que sea preciso (construid módelos para media y varianza simultáneamente.

Iter: 1 fn: -323.9728    Pars:  0.00600054 0.00002065 0.41064643 0.58835341
Iter: 2 fn: -323.9728    Pars:  0.00600044 0.00002065 0.41063341 0.58836659
solnp--> Completed in 2 iterations

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.006000    0.001377   4.3585 0.000013
omega   0.000021    0.000013   1.6444 0.100099
alpha1  0.410633    0.143311   2.8653 0.004166
beta1   0.588367    0.094845   6.2035 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu      0.006000    0.002302   2.6068 0.009138
omega   0.000021    0.000017   1.2101 0.226241
alpha1  0.410633    0.218250   1.8815 0.059907
beta1   0.588367    0.134414   4.3773 0.000012

LogLikelihood : 323.9728 

Information Criteria
------------------------------------
                    
Akaike       -5.5647
Bayes        -5.4693
Shibata      -5.5671
Hannan-Quinn -5.5260

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic  p-value
Lag[1]                      7.498 0.006177
Lag[2*(p+q)+(p+q)-1][2]     7.974 0.006573
Lag[4*(p+q)+(p+q)-1][5]    11.160 0.004617
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                    0.03843  0.8446
Lag[2*(p+q)+(p+q)-1][5]   2.29488  0.5506
Lag[4*(p+q)+(p+q)-1][9]   3.74639  0.6321
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]   0.01708 0.500 2.000  0.8960
ARCH Lag[5]   3.18781 1.440 1.667  0.2637
ARCH Lag[7]   3.67295 2.315 1.543  0.3962

Nyblom stability test
------------------------------------
Joint Statistic:  1.8309
Individual Statistics:             
mu     0.8683
omega  0.7110
alpha1 0.2063
beta1  0.5113

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.6085 0.5441    
Negative Sign Bias  0.2452 0.8067    
Positive Sign Bias  0.4985 0.6191    
Joint Effect        1.8150 0.6117    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     33.70     0.019956
2    30     49.09     0.011306
3    40     65.52     0.004946
4    50     65.43     0.058230


Elapsed time : 1.084001 

Apartado 3

Realizad predicciones de media y varianza.

mx.garch11.fcst <- ugarchforecast(mx.garch11.fit, n.ahead=40)
plot(mx.garch11.fcst, which=1)

plot(mx.garch11.fcst, which=3)

Tipo de interés a largo plazo

Apartado 1

Comprobad si hay efectos ARCH (gráficos y contrastes LM).

Si ninguna de las variables presenta efectos ARCH usad datos de precios de algún activo financiero o índice bursátil.

lt.ret <- CalculateReturns(lt, method="log")
lt.ret <- lt.ret[-1,]
dataToPlot <- cbind(lt.ret, lt.ret^2, abs(lt.ret))
colnames(dataToPlot) <- c("Returns", "Returns^2", "abs(Returns)")
plot.zoo(dataToPlot, main="Long Term Interest Rate Monthly Returns", col="blue")
par(mfrow=c(3,1))

acf(lt.ret, main="Long Term Interest Rate Returns")
acf(lt.ret^2, main="Long Term Interest Rate Returns^2")

acf(abs(lt.ret), main="Long Term Interest Rate abs(Returns)")
par(mfrow=c(1,1)) 

table.Stats(lt.ret)

Box.test(coredata(lt.ret^2), type="Ljung-Box", lag = 12)

    Box-Ljung test

data:  coredata(lt.ret^2)
X-squared = 38.574, df = 12, p-value = 0.0001237

ArchTest(lt.ret)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  lt.ret
Chi-squared = 32.78, df = 12, p-value = 0.001048

En los gráficos de retornos se ven varios periodos donde aumentan los retornos considerablemente y otros en los que se mantienen más bajos por lo que podría decirse que la varianza no es constante y hay efectos ARCH.

Se rechaza la hipótesis nula del LM-test y por tanto nuestra serie tiene efectos ARCH.

Apartado 2

Construid modelos de la familia GARCH para las variables que sea preciso (construid módelos para media y varianza simultáneamente.

Iter: 1 fn: -286.4082    Pars:  -0.0023904  0.0001566  0.1625106  0.7996552
Iter: 2 fn: -286.4082    Pars:  -0.0023906  0.0001566  0.1625133  0.7996524
solnp--> Completed in 2 iterations

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.002391    0.003390 -0.70514 0.480721
omega   0.000157    0.000071  2.21574 0.026709
alpha1  0.162513    0.054125  3.00256 0.002677
beta1   0.799652    0.048439 16.50841 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.002391    0.003785 -0.63158 0.527660
omega   0.000157    0.000092  1.70259 0.088646
alpha1  0.162513    0.069349  2.34343 0.019107
beta1   0.799652    0.043740 18.28194 0.000000

LogLikelihood : 286.4082 

Information Criteria
------------------------------------
                    
Akaike       -3.0864
Bayes        -3.0163
Shibata      -3.0874
Hannan-Quinn -3.0580

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.3949  0.5297
Lag[2*(p+q)+(p+q)-1][2]    0.6263  0.6372
Lag[4*(p+q)+(p+q)-1][5]    2.3821  0.5313
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.6125  0.4338
Lag[2*(p+q)+(p+q)-1][5]    1.2153  0.8093
Lag[4*(p+q)+(p+q)-1][9]    3.2282  0.7208
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]  0.005119 0.500 2.000  0.9430
ARCH Lag[5]  0.366499 1.440 1.667  0.9221
ARCH Lag[7]  0.596393 2.315 1.543  0.9688

Nyblom stability test
------------------------------------
Joint Statistic:  1.1239
Individual Statistics:              
mu     0.22448
omega  0.20742
alpha1 0.07327
beta1  0.20602

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.8519 0.3954    
Negative Sign Bias  0.0192 0.9847    
Positive Sign Bias  0.2936 0.7694    
Joint Effect        1.7647 0.6226    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     43.12     0.001248
2    30     54.87     0.002566
3    40     64.21     0.006692
4    50     80.11     0.003316


Elapsed time : 0.382998 

Apartado 3

Realizad predicciones de media y varianza.

lt.garch11.fcst <- ugarchforecast(lt.garch11.fit, n.ahead=36)
plot(lt.garch11.fcst, which=1)

plot(lt.garch11.fcst, which=3)

Tipo de interés a medio plazo

Apartado 1

Comprobad si hay efectos ARCH (gráficos y contrastes LM).

Si ninguna de las variables presenta efectos ARCH usad datos de precios de algún activo financiero o índice bursátil.

mt.ret <- CalculateReturns(mt, method="log")
mt.ret <- mt.ret[-1,]
dataToPlot <- cbind(mt.ret, mt.ret^2, abs(mt.ret))
colnames(dataToPlot) <- c("Returns", "Returns^2", "abs(Returns)")
plot.zoo(dataToPlot, main="Medium Term Interest Rate Monthly Returns", col="blue")
par(mfrow=c(3,1))

acf(mt.ret, main="Medium Term Interest Rate Returns")
acf(mt.ret^2, main="Medium Term Interest Rate Returns^2")

acf(abs(mt.ret), main="Medium Term Interest Rate abs(Returns)")
par(mfrow=c(1,1)) 

table.Stats(mt.ret)

Box.test(coredata(mt.ret^2), type="Ljung-Box", lag = 12)

    Box-Ljung test

data:  coredata(mt.ret^2)
X-squared = 73.113, df = 12, p-value = 8.34e-11

ArchTest(mt.ret)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  mt.ret
Chi-squared = 47.132, df = 12, p-value = 4.422e-06

En los gráficos de retornos se ven varios periodos donde aumentan los retornos considerablemente y otros en los que se mantienen más bajos por lo que podría decirse que la varianza no es constante y hay efectos ARCH.

Se rechaza la hipótesis nula del LM-test y por tanto nuestra serie tiene efectos ARCH.

Apartado 2

Construid modelos de la familia GARCH para las variables que sea preciso (construid módelos para media y varianza simumtáneamente.

Iter: 1 fn: -356.9919    Pars:  -0.0024712  0.0001027  0.3939694  0.6050306
Iter: 2 fn: -356.9919    Pars:  -0.0024712  0.0001027  0.3939692  0.6050307
solnp--> Completed in 2 iterations

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.002471    0.001914  -1.2909 0.196749
omega   0.000103    0.000040   2.5391 0.011115
alpha1  0.393969    0.100570   3.9174 0.000090
beta1   0.605031    0.078667   7.6910 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.002471    0.002530 -0.97674 0.328697
omega   0.000103    0.000097  1.05464 0.291590
alpha1  0.393969    0.152211  2.58832 0.009645
beta1   0.605031    0.148988  4.06095 0.000049

LogLikelihood : 356.9919 

Information Criteria
------------------------------------
                    
Akaike       -3.8578
Bayes        -3.7877
Shibata      -3.8588
Hannan-Quinn -3.8294

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic   p-value
Lag[1]                      15.45 8.482e-05
Lag[2*(p+q)+(p+q)-1][2]     16.73 3.214e-05
Lag[4*(p+q)+(p+q)-1][5]     23.03 2.817e-06
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                   0.001927  0.9650
Lag[2*(p+q)+(p+q)-1][5]  0.392547  0.9729
Lag[4*(p+q)+(p+q)-1][9]  0.824442  0.9924
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]  0.004622 0.500 2.000  0.9458
ARCH Lag[5]  0.459258 1.440 1.667  0.8956
ARCH Lag[7]  0.601562 2.315 1.543  0.9683

Nyblom stability test
------------------------------------
Joint Statistic:  0.6874
Individual Statistics:              
mu     0.12716
omega  0.11603
alpha1 0.12883
beta1  0.09834

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           0.4454 0.6566    
Negative Sign Bias  0.1423 0.8870    
Positive Sign Bias  0.2333 0.8158    
Joint Effect        0.6297 0.8896    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     138.4    3.657e-20
2    30     173.6    1.549e-22
3    40     187.9    2.201e-21
4    50     211.8    3.972e-22


Elapsed time : 0.471998 

Apartado 3

Realizad predicciones de media y varianza.

mt.garch11.fcst <- ugarchforecast(mt.garch11.fit, n.ahead=36)
plot(mt.garch11.fcst, which=1)

plot(mt.garch11.fcst, which=3)

Tipo de interés a corto plazo

Apartado 1

Comprobad si hay efectos ARCH (gráficos y contrastes LM).

Si ninguna de las variables presenta efectos ARCH usad datos de precios de algún activo financiero o índice bursátil.

st.ret <- CalculateReturns(st, method="log")
st.ret <- st.ret[-1,]
dataToPlot <- cbind(st.ret, st.ret^2, abs(st.ret))
colnames(dataToPlot) <- c("Returns", "Returns^2", "abs(Returns)")
plot.zoo(dataToPlot, main="Short Term Interest Rate Monthly Returns", col="blue")
par(mfrow=c(3,1))

acf(st.ret, main="Short Term Interest Rate Returns")
acf(st.ret^2, main="Short Term Interest Rate Returns^2")

acf(abs(st.ret), main="Short Term Interest Rate abs(Returns)")
par(mfrow=c(1,1)) 

table.Stats(st.ret)

Box.test(coredata(st.ret^2), type="Ljung-Box", lag = 12)

    Box-Ljung test

data:  coredata(st.ret^2)
X-squared = 35.965, df = 12, p-value = 0.0003282

ArchTest(st.ret)

    ARCH LM-test; Null hypothesis: no ARCH effects

data:  st.ret
Chi-squared = 29.914, df = 12, p-value = 0.002877

En los gráficos de retornos se ven varios periodos donde aumentan los retornos y otros en los que se mantienen más bajos por lo que podría decirse que la varianza no es constante y hay efectos ARCH.

Se rechaza la hipótesis nula del LM-test y por tanto nuestra serie tiene efectos ARCH.

Apartado 2

Construid modelos de la familia GARCH para las variables que sea preciso (construid módelos para media y varianza simustáneamente.

Iter: 1 fn: -368.0697    Pars:  -0.00093621  0.00005858  0.46359929  0.53540063
Iter: 2 fn: -368.0697    Pars:  -0.00093629  0.00005857  0.46359739  0.53540256
solnp--> Completed in 2 iterations

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics   
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : norm 

Optimal Parameters
------------------------------------
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000936    0.001638 -0.57173 0.567502
omega   0.000059    0.000031  1.91862 0.055033
alpha1  0.463597    0.102036  4.54345 0.000006
beta1   0.535403    0.085346  6.27330 0.000000

Robust Standard Errors:
        Estimate  Std. Error  t value Pr(>|t|)
mu     -0.000936    0.002105 -0.44482 0.656449
omega   0.000059    0.000042  1.38240 0.166850
alpha1  0.463597    0.129675  3.57507 0.000350
beta1   0.535403    0.132924  4.02789 0.000056

LogLikelihood : 368.0697 

Information Criteria
------------------------------------
                    
Akaike       -3.9789
Bayes        -3.9088
Shibata      -3.9798
Hannan-Quinn -3.9505

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic   p-value
Lag[1]                      17.38 3.062e-05
Lag[2*(p+q)+(p+q)-1][2]     18.93 8.510e-06
Lag[4*(p+q)+(p+q)-1][5]     28.29 9.598e-08
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.4363  0.5089
Lag[2*(p+q)+(p+q)-1][5]    1.8887  0.6450
Lag[4*(p+q)+(p+q)-1][9]    2.6319  0.8178
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]     1.744 0.500 2.000  0.1867
ARCH Lag[5]     2.552 1.440 1.667  0.3617
ARCH Lag[7]     2.703 2.315 1.543  0.5714

Nyblom stability test
------------------------------------
Joint Statistic:  1.8368
Individual Statistics:             
mu     0.2574
omega  0.2570
alpha1 0.6818
beta1  0.4567

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.07 1.24 1.6
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------
                   t-value   prob sig
Sign Bias           1.0191 0.3095    
Negative Sign Bias  0.2295 0.8187    
Positive Sign Bias  0.1138 0.9095    
Joint Effect        1.6913 0.6389    


Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     20.61       0.3589
2    30     34.54       0.2200
3    40     38.42       0.4961
4    50     51.15       0.3891


Elapsed time : 0.608001 

Apartado 3

Realizad predicciones de media y varianza.

st.garch11.fcst <- ugarchforecast(st.garch11.fit, n.ahead=36)
plot(st.garch11.fcst, which=1)

plot(st.garch11.fcst, which=3)