# Using pspatreg with spatial panel data

#### Román Mínguez (UCLM), Roberto Basile (UNIVAQ), María Durbán (UC3M)

#### 2023-10-06

    #library(pspatreg)
    devtools::load_all()
    library(spatialreg)
    library(spdep)
    library(sf)
    library(plm)
    library(ggplot2)
    library(dplyr)
    library(splm)

# 1 Models for spatial panel data

This section focuses on the semiparametric P-Spline model for spatial panel data. The model may include a smooth spatio-temporal trend, a spatial lag of dependent and independent variables, a time lag of the dependent variable and of its spatial lag, and a time series autoregressive noise. Specifically, we consider a spatio-temporal ANOVA model, disaggregating the trend into spatial and temporal main effects, as well as second- and third-order interactions between them.

The empirical illustration is based on data on regional unemployment in Italy. This example shows that this model represents a valid alternative to parametric methods aimed at disentangling strong and weak cross-sectional dependence when both spatial and temporal heterogeneity are smoothly distributed (see Mı́nguez, Basile, and Durbán 2020). The section is organized as follows:

- Description of dataset, spatial weights matrix and model specifications;

- Estimation results of linear spatial models and comparison with the results obtained with splm;

- Estimation results of semiparametric spatial models.

## 1.1 Dataset, spatial weights matrix and model specifications

The package provides the panel data `unemp_it` (an object of class `data.frame`) and the spatial weights matrix `Wsp_it` (a 103 by 103 square matrix). The raw data - a balanced panel with 103 Italian provinces observed for each year between 1996 and 2019 - can be transformed in a spatial polygonal dataset of class `sf` after having joined the `data.frame` object with the shapefile of Italian provinces:

    data(unemp_it, package = "pspatreg")
    unemp_it_sf <- st_as_sf(dplyr::left_join(unemp_it, map_it, by = c("prov" = "COD_PRO")))

The matrix `Wsp_it` is a standardized inverse distance W matrix. Using `spdep` we transform it in a list of neighbors object:

    lwsp_it <- spdep::mat2listw(Wsp_it)
    summary(lwsp_it)

    ## Characteristics of weights list object:
    ## Neighbour list object:
    ## Number of regions: 103
    ## Number of nonzero links: 434
    ## Percentage nonzero weights: 4.090866
    ## Average number of links: 4.213592
    ## Link number distribution:
    ##
    ##  1  2  3  4  5  6  7  8  9
    ##  7 20 15 16 17 11 10  6  1
    ## 7 least connected regions:
    ## 32 75 78 80 81 90 92 with 1 link
    ## 1 most connected region:
    ## 15 with 9 links
    ##
    ## Weights style: M
    ## Weights constants summary:
    ##     n    nn  S0       S1       S2
    ## M 103 10609 103 74.35526 431.5459

## 1.2 Linear model (comparison with splm)

Using these data, we first estimate fully parametric spatial linear autoregressive panel models using the function `pspatfit()` included in the package pspatreg (in the default based on the REML estimator) and compare them with the results obtained using the functions provided by the package splm (based on the ML estimator).

### 1.2.1 Spatial Lag model (SAR). REML estimates using `pspatfit()`

We consider here a fixed effects specification, including both fixed spatial and time effects:

\[y_{it}=\rho \sum_{j=1}^N w_{ij,N} y_{jt} + \sum_{k=1}^K \beta_k x_{k,it}+ \alpha_i+\theta_t+\epsilon_{it}\]

\[\epsilon_{it} \sim i.i.d.(0,\sigma^2_\epsilon)\]

    formlin <- unrate ~ empgrowth + partrate + agri + cons + serv

    Linear_WITHIN_sar_REML <- pspatfit(formula = formlin,
                       data = unemp_it,
                       listw = lwsp_it,
                       demean = TRUE,
                       eff_demean = "twoways",
                       type = "sar",
                       index = c("prov", "year"))

    summary(Linear_WITHIN_sar_REML)

    ##
    ##  Call
    ## pspatfit(formula = formlin, data = unemp_it, listw = lwsp_it,
    ##     type = "sar", demean = TRUE, eff_demean = "twoways", index = c("prov",
    ##         "year"))
    ##
    ##  Parametric Terms
    ##             Estimate Std. Error t value  Pr(>|t|)
    ## empgrowth -0.129739   0.014091 -9.2074 < 2.2e-16 ***
    ## partrate   0.393087   0.023315 16.8595 < 2.2e-16 ***
    ## agri      -0.036052   0.027267 -1.3222 0.1862167
    ## cons      -0.166196   0.044510 -3.7339 0.0001928 ***
    ## serv       0.012378   0.020597  0.6009 0.5479300
    ## rho        0.265671   0.018858 14.0880 < 2.2e-16 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total:      6
    ##  Sigma: 1.86929
    ##  AIC:  5396.53
    ##  BIC:  5431.41

    Linear_WITHIN_sar_ML <- spml(formlin,
                   data = unemp_it,
                   index=c("prov","year"),
                   listw = lwsp_it,
                   model="within",
                   effect = "twoways",
                   spatial.error="none",
                   lag=TRUE,
                   Hess = FALSE)

    round(data.frame(Linear_WITHIN_sar_REML = c(Linear_WITHIN_sar_REML$rho,
                                                Linear_WITHIN_sar_REML$bfixed),
                    Linear_WITHIN_sar_ML = c(Linear_WITHIN_sar_ML$coefficients[1],
                                             Linear_WITHIN_sar_ML$coefficients[-1])),3)

    ##                 Linear_WITHIN_sar_REML Linear_WITHIN_sar_ML
    ## rho                              0.266                0.266
    ## fixed_empgrowth                 -0.130               -0.130
    ## fixed_partrate                   0.393                0.392
    ## fixed_agri                      -0.036               -0.037
    ## fixed_cons                      -0.166               -0.167
    ## fixed_serv                       0.012                0.012

Clearly, both methods give exactly the same results, at least at the third digit level.

Extract coefficients:

    coef(Linear_WITHIN_sar_REML)

    ##         rho   empgrowth    partrate        agri        cons
    ##  0.26567078 -0.12973894  0.39308715 -0.03605243 -0.16619608
    ##        serv
    ##  0.01237770

Extract fitted values and residuals:

    fits <- fitted(Linear_WITHIN_sar_REML)
    resids <- residuals(Linear_WITHIN_sar_REML)

Extract log-likelihood and restricted log-likelihhod:

    logLik(Linear_WITHIN_sar_REML)

    ## 'log Lik.' -2692.266 (df=6)

    logLik(Linear_WITHIN_sar_REML, REML = TRUE)

    ## 'log Lik.' -2711.112 (df=6)

Extract the covariance matrix of estimated coefficients. Argument `bayesian` allows to get bayesian (default) or frequentist covariances:

    vcov(Linear_WITHIN_sar_REML)

    ##               empgrowth      partrate          agri          cons
    ## empgrowth  1.985464e-04 -8.106848e-05 -3.050359e-06  3.032827e-05
    ## partrate  -8.106848e-05  5.436097e-04 -7.233434e-05 -1.649890e-04
    ## agri      -3.050359e-06 -7.233434e-05  7.434641e-04  1.891487e-04
    ## cons       3.032827e-05 -1.649890e-04  1.891487e-04  1.981149e-03
    ## serv      -6.861998e-06 -7.891185e-05  2.741340e-04  2.432964e-04
    ##                    serv
    ## empgrowth -6.861998e-06
    ## partrate  -7.891185e-05
    ## agri       2.741340e-04
    ## cons       2.432964e-04
    ## serv       4.242353e-04

    vcov(Linear_WITHIN_sar_REML, bayesian = FALSE)

    ##               empgrowth      partrate          agri          cons
    ## empgrowth  1.985464e-04 -8.106848e-05 -3.050359e-06  3.032827e-05
    ## partrate  -8.106848e-05  5.436097e-04 -7.233434e-05 -1.649890e-04
    ## agri      -3.050359e-06 -7.233434e-05  7.434641e-04  1.891487e-04
    ## cons       3.032827e-05 -1.649890e-04  1.891487e-04  1.981149e-03
    ## serv      -6.861998e-06 -7.891185e-05  2.741340e-04  2.432964e-04
    ##                    serv
    ## empgrowth -6.861998e-06
    ## partrate  -7.891185e-05
    ## agri       2.741340e-04
    ## cons       2.432964e-04
    ## serv       4.242353e-04

A print method to get printed coefficients, standard errors and p-values of parametric terms:

    print(Linear_WITHIN_sar_REML)

    ##           Estimate Std. Error t value Pr(>|t|)
    ## empgrowth  -0.1297     0.0141 -9.2074   0.0000
    ## partrate    0.3931     0.0233 16.8595   0.0000
    ## agri       -0.0361     0.0273 -1.3222   0.1862
    ## cons       -0.1662     0.0445 -3.7339   0.0002
    ## serv        0.0124     0.0206  0.6009   0.5479
    ## rho         0.2657     0.0189 14.0880   0.0000

    summary(Linear_WITHIN_sar_REML)

    ##
    ##  Call
    ## pspatfit(formula = formlin, data = unemp_it, listw = lwsp_it,
    ##     type = "sar", demean = TRUE, eff_demean = "twoways", index = c("prov",
    ##         "year"))
    ##
    ##  Parametric Terms
    ##             Estimate Std. Error t value  Pr(>|t|)
    ## empgrowth -0.129739   0.014091 -9.2074 < 2.2e-16 ***
    ## partrate   0.393087   0.023315 16.8595 < 2.2e-16 ***
    ## agri      -0.036052   0.027267 -1.3222 0.1862167
    ## cons      -0.166196   0.044510 -3.7339 0.0001928 ***
    ## serv       0.012378   0.020597  0.6009 0.5479300
    ## rho        0.265671   0.018858 14.0880 < 2.2e-16 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total:      6
    ##  Sigma: 1.86929
    ##  AIC:  5396.53
    ##  BIC:  5431.41

    summary(Linear_WITHIN_sar_ML)

    ## Spatial panel fixed effects lag model
    ##
    ##
    ## Call:
    ## spml(formula = formlin, data = unemp_it, index = c("prov", "year"),
    ##     listw = lwsp_it, model = "within", effect = "twoways", lag = TRUE,
    ##     spatial.error = "none", Hess = FALSE)
    ##
    ## Residuals:
    ##      Min.   1st Qu.    Median   3rd Qu.      Max.
    ## -8.045700 -1.068404 -0.035768  1.014227  7.816307
    ##
    ## Spatial autoregressive coefficient:
    ##        Estimate Std. Error t-value  Pr(>|t|)
    ## lambda 0.266004   0.020636   12.89 < 2.2e-16 ***
    ##
    ## Coefficients:
    ##            Estimate Std. Error t-value  Pr(>|t|)
    ## empgrowth -0.129530   0.014078 -9.2009 < 2.2e-16 ***
    ## partrate   0.391597   0.023464 16.6894 < 2.2e-16 ***
    ## agri      -0.036771   0.027219 -1.3510 0.1767102
    ## cons      -0.166896   0.044420 -3.7572 0.0001718 ***
    ## serv       0.012191   0.020559  0.5930 0.5532105
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Computing average direct, indirect and total marginal impacts:

    imp_parvar_sar <- impactspar(Linear_WITHIN_sar_REML, listw = lwsp_it)
    summary(imp_parvar_sar)

    ##
    ##  Total Parametric Impacts (sar)
    ##            Estimate Std. Error   t value Pr(>|t|)
    ## empgrowth -0.175364   0.020100 -8.724637   0.0000
    ## partrate   0.534558   0.032988 16.204584   0.0000
    ## agri      -0.047202   0.037812 -1.248356   0.2119
    ## cons      -0.228481   0.058921 -3.877720   0.0001
    ## serv       0.017354   0.027066  0.641171   0.5214
    ##
    ##  Direct Parametric Impacts (sar)
    ##            Estimate Std. Error   t value Pr(>|t|)
    ## empgrowth -0.131815   0.014742 -8.941567   0.0000
    ## partrate   0.401828   0.022938 17.518404   0.0000
    ## agri      -0.035503   0.028408 -1.249771   0.2114
    ## cons      -0.171776   0.044182 -3.887906   0.0001
    ## serv       0.013036   0.020332  0.641190   0.5214
    ##
    ##  Indirect Parametric Impacts (sar)
    ##             Estimate Std. Error    t value Pr(>|t|)
    ## empgrowth -0.0435494  0.0062928 -6.9205084   0.0000
    ## partrate   0.1327297  0.0140779  9.4282579   0.0000
    ## agri      -0.0116994  0.0094721 -1.2351363   0.2168
    ## cons      -0.0567047  0.0153916 -3.6841194   0.0002
    ## serv       0.0043173  0.0067593  0.6387246   0.5230

### 1.2.2 Spatial error within model (SEM). REML estimates using `pspatfit()`:

\[y_{it}= \sum_{k=1}^K \beta_k x_{k,it}+\alpha_i+\theta_t+ \epsilon_{it}\]

\[\epsilon_{it}=\theta \sum_{j=1}^N w_{ij,N}\epsilon_{it}+u_{it}\]

\[u_{it} \sim i.i.d.(0,\sigma^2_u)\]

    Linear_WITHIN_sem_REML <- pspatfit(formlin,
                                   data = unemp_it,
                                   demean = TRUE,
                                   eff_demean = "twoways",
                                   listw = lwsp_it,
                                   index = c("prov", "year"),
                                   type = "sem")

    Linear_WITHIN_sem_ML <- spml(formlin,
                             data = unemp_it,
                             index=c("prov","year"),
                             listw = lwsp_it,
                             model="within",
                             effect = "twoways",
                             spatial.error="b",
                             lag=FALSE,
                             Hess = FALSE)

    round(data.frame(Linear_WITHIN_sem_REML = c(Linear_WITHIN_sem_REML$delta,
                                                Linear_WITHIN_sem_REML$bfixed),
                     Linear_WITHIN_sem_ML = c(Linear_WITHIN_sem_ML$spat.coef,
                                              Linear_WITHIN_sem_ML$coefficients[-1])),3)

    ##                 Linear_WITHIN_sem_REML Linear_WITHIN_sem_ML
    ## delta                            0.283                0.283
    ## fixed_empgrowth                 -0.134               -0.134
    ## fixed_partrate                   0.400                0.399
    ## fixed_agri                      -0.032               -0.033
    ## fixed_cons                      -0.186               -0.188
    ## fixed_serv                       0.030                0.031

## 1.3 Semiparametric model without spatial trends

Now, we estimate an additive semiparametric model with three parametric linear terms (for `partrate`, `agri`, and `cons`) and two nonparametric smooth terms (for `serv` and `empgrowth`), but without including any control for spatial and temporal autocorrelation and for the spatio-temporal heterogeneity: \[y_{it}= \sum_{k=1}^K \beta_k z_{k,it} + \sum_{\delta=1}^{\Delta} g_\delta(x_{\delta_{it}}) + \epsilon_{it}\]

\[\epsilon_{it} \sim i.i.d.(0,\sigma^2_\epsilon)\]

    formgam <- unrate ~ partrate + agri + cons +
                      pspl(serv, nknots = 15) +
                      pspl(empgrowth, nknots = 20)

    gam <- pspatfit(formgam, data = unemp_it)
    summary(gam)

    ##
    ##  Call
    ## pspatfit(formula = formgam, data = unemp_it)
    ##
    ##  Parametric Terms
    ##                   Estimate Std. Error  t value  Pr(>|t|)
    ## (Intercept)     24.423844   1.095796  22.2887 < 2.2e-16 ***
    ## partrate        -0.370101   0.017980 -20.5837 < 2.2e-16 ***
    ## agri             0.346768   0.020121  17.2342 < 2.2e-16 ***
    ## cons            -0.185432   0.055401  -3.3471 0.0008289 ***
    ## pspl(serv)       1.921073   0.283973   6.7650 1.660e-11 ***
    ## pspl(empgrowth) -0.432407   0.088512  -4.8853 1.099e-06 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Non-Parametric Terms
    ##                     EDF
    ## pspl(serv)      5.4818
    ## pspl(empgrowth) 0.2496
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total: 11.7314
    ##  Sigma: 4.07483
    ##  AIC:  9456.09
    ##  BIC:  9524.28

The same model, but with a spatial autoregressive term (SAR): \[y_{it}= \rho \sum_{j=1}^N w_{ij,N} y_{jt} +\sum_{k=1}^K \beta_k z_{k,it} + \sum_{\delta=1}^{\Delta} g_\delta(x_{\delta_{it}}) + \epsilon_{it}\]

\[\epsilon_{it} \sim i.i.d.(0,\sigma^2_\epsilon)\]

    gamsar <- pspatfit(formgam, data = unemp_it, listw = lwsp_it, method = "eigen", type = "sar")
    summary(gamsar)

    ##
    ##  Call
    ## pspatfit(formula = formgam, data = unemp_it, listw = lwsp_it,
    ##     type = "sar", method = "eigen")
    ##
    ##  Parametric Terms
    ##                   Estimate Std. Error  t value  Pr(>|t|)
    ## (Intercept)     10.068382   0.725355  13.8806 < 2.2e-16 ***
    ## partrate        -0.162431   0.011705 -13.8768 < 2.2e-16 ***
    ## agri             0.093943   0.013111   7.1651 1.023e-12 ***
    ## cons            -0.095438   0.036074  -2.6456  0.008207 **
    ## pspl(serv)       0.598250   0.216404   2.7645  0.005743 **
    ## pspl(empgrowth) -0.218982   0.054368  -4.0278 5.802e-05 ***
    ## rho              0.658979   0.011261  58.5211 < 2.2e-16 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Non-Parametric Terms
    ##                     EDF
    ## pspl(serv)      6.9167
    ## pspl(empgrowth) 0.0000
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total: 13.9168
    ##  Sigma: 3.56329
    ##  AIC:  7763.26
    ##  BIC:  7844.15

and a spatial error term: \[y_{it}= \sum_{k=1}^K \beta_k z_{k,it} + \sum_{\delta=1}^{\Delta} g_\delta(x_{\delta_{it}}) + \epsilon_{it}\]

\[\epsilon_{it} = \delta \sum_{j=1}^N w_{ij,N}\epsilon_{it}+u_{it}\]

\[u_{it} \sim i.i.d.(0,\sigma^2_u)\]

    gamsem <- pspatfit(formgam, data = unemp_it, listw = lwsp_it, method = "eigen", type = "sem")

    ## Error in .solve.checkCond(a, tol) :
    ##   'a' is computationally singular, rcond(a)=4.30414e-31
    ## Error in .solve.checkCond(a, tol) :
    ##   'a' is computationally singular, rcond(a)=7.36531e-32
    ## Error in .solve.checkCond(a, tol) :
    ##   'a' is computationally singular, rcond(a)=6.93286e-32

    summary(gamsem)

    ##
    ##  Call
    ## pspatfit(formula = formgam, data = unemp_it, listw = lwsp_it,
    ##     type = "sem", method = "eigen")
    ##
    ##  Parametric Terms
    ##                   Estimate Std. Error  t value  Pr(>|t|)
    ## (Intercept)     19.397175   1.059863  18.3016 < 2.2e-16 ***
    ## partrate        -0.217678   0.020990 -10.3704 < 2.2e-16 ***
    ## agri             0.020614   0.015061   1.3687  0.171223
    ## cons            -0.087933   0.039020  -2.2535  0.024314 *
    ## pspl(serv)       0.544096   0.269666   2.0177  0.043734 *
    ## pspl(empgrowth) -0.142984   0.055169  -2.5917  0.009606 **
    ## delta            0.751250   0.010998  68.3083 < 2.2e-16 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Non-Parametric Terms
    ##                     EDF
    ## pspl(serv)      9.8356
    ## pspl(empgrowth) 0.0000
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total: 16.8356
    ##  Sigma: 4.86032
    ##  AIC:  8082.93
    ##  BIC:  8180.8

We can control for spatio-temporal heterogeneity by including a PS-ANOVA spatial trend in 3d. The interaction terms (`f12`,`f1t`,`f2t` and `f12t`) with nested basis. Remark: `nest_sp1`, `nest_sp2` and `nest_time` must be divisors of `nknots`.

\[y_{it}= \sum_{k=1}^K \beta_k z_{k,it} + \sum_{\delta=1}^{\Delta} g_\delta(x_{\delta_{it}}) + f_1(s_{1i})+f_2(s_{2i})+f_{\tau}(\tau_t)+ \\ f_{1,2}(s_{1i},s_{2i})+f_{1,\tau}(s_{1i},\tau_t)+f_{2,\tau}+(s_{2i},\tau_t)+f_{1,2,\tau}(s_{1i},s_{2i},\tau_t)+\epsilon_{it}\]

\[\epsilon_{it} \sim i.i.d.(0,\sigma^2_\epsilon)\]

    form3d_psanova <- unrate ~ partrate + agri + cons +
                      pspl(serv, nknots = 15) +
                      pspl(empgrowth, nknots = 20) +
                      pspt(long, lat, year,
                           nknots = c(18,18,8), psanova = TRUE,
                           nest_sp1 = c(1, 2, 3),
                           nest_sp2 = c(1, 2, 3),
                           nest_time = c(1, 2, 2))

    sp3danova <- pspatfit(form3d_psanova, data = unemp_it,
                        listw = lwsp_it, method = "Chebyshev")

    summary(sp3danova)

    ##
    ##  Call
    ## pspatfit(formula = form3d_psanova, data = unemp_it, listw = lwsp_it,
    ##     method = "Chebyshev")
    ##
    ##  Parametric Terms
    ##                    Estimate Std. Error t value  Pr(>|t|)
    ## (Intercept)       3.975659   8.445688  0.4707    0.6379
    ## partrate          0.149545   0.021941  6.8158 1.197e-11 ***
    ## agri             -0.020505   0.019754 -1.0380    0.2994
    ## cons             -0.038684   0.040441 -0.9566    0.3389
    ## f1_main.1        -1.839321  13.263307 -0.1387    0.8897
    ## f2_main.1       -15.544083  10.965999 -1.4175    0.1565
    ## ft_main.1         2.414483   4.895533  0.4932    0.6219
    ## f12_int.1        -8.798610  12.357416 -0.7120    0.4765
    ## f1t_int.1         4.110521   6.471417  0.6352    0.5254
    ## f2t_int.1        -2.120851   6.780980 -0.3128    0.7545
    ## f12t_int.1        1.412301   7.659432  0.1844    0.8537
    ## pspl(serv)        0.091533   0.168186  0.5442    0.5863
    ## pspl(empgrowth)  -0.287978   0.065553 -4.3931 1.169e-05 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Non-Parametric Terms
    ##                     EDF
    ## pspl(serv)      5.6815
    ## pspl(empgrowth) 2.6263
    ##
    ##  Non-Parametric Spatio-Temporal Trend
    ##         EDF
    ## f1   10.667
    ## f2    9.733
    ## ft    7.782
    ## f12  45.424
    ## f1t   4.239
    ## f2t  21.473
    ## f12t 82.111
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total: 202.737
    ##  Sigma: 1.53943
    ##  AIC:  5975.89
    ##  BIC:  7154.35

A semiparametric model with a PS-ANOVA spatial trend in 3d with the exclusion of some ANOVA components

    form3d_psanova_restr <- unrate ~ partrate + agri + cons +
                            pspl(serv, nknots = 15) +
                            pspl(empgrowth, nknots = 20) +
                            pspt(long, lat, year,
                             nknots = c(18,18,8), psanova = TRUE,
                             nest_sp1 = c(1, 2, 3),
                             nest_sp2 = c(1, 2, 3),
                             nest_time = c(1, 2, 2),
                             f1t = FALSE, f2t = FALSE)

    sp3danova_restr <- pspatfit(form3d_psanova_restr, data = unemp_it,
                        listw = lwsp_it, method = "Chebyshev")

    summary(sp3danova_restr)

    ##
    ##  Call
    ## pspatfit(formula = form3d_psanova_restr, data = unemp_it, listw = lwsp_it,
    ##     method = "Chebyshev")
    ##
    ##  Parametric Terms
    ##                    Estimate Std. Error t value  Pr(>|t|)
    ## (Intercept)       4.366141   8.446761  0.5169   0.60528
    ## partrate          0.150986   0.021906  6.8923 7.089e-12 ***
    ## agri             -0.027583   0.019676 -1.4018   0.16110
    ## cons             -0.022771   0.040534 -0.5618   0.57432
    ## f1_main.1        -1.277158  13.265175 -0.0963   0.92331
    ## f2_main.1       -15.939808  10.969054 -1.4532   0.14632
    ## ft_main.1         4.182334   1.888331  2.2148   0.02687 *
    ## f12_int.1        -8.219153  12.404068 -0.6626   0.50764
    ## f12t_int.1       14.512194   6.281550  2.3103   0.02096 *
    ## pspl(serv)        0.064638   0.162365  0.3981   0.69059
    ## pspl(empgrowth)  -0.297971   0.065505 -4.5488 5.680e-06 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Non-Parametric Terms
    ##                     EDF
    ## pspl(serv)      5.4063
    ## pspl(empgrowth) 2.6206
    ##
    ##  Non-Parametric Spatio-Temporal Trend
    ##          EDF
    ## f1    10.591
    ## f2     9.663
    ## ft     7.765
    ## f12   45.581
    ## f1t    0.000
    ## f2t    0.000
    ## f12t 114.210
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total: 206.837
    ##  Sigma: 1.5383
    ##  AIC:  6019.34
    ##  BIC:  7221.64

Now we add a spatial lag (sar) and temporal correlation in the noise of PSANOVA 3d model.

\[y_{it}= \rho \sum_{j=1}^N w_{ij,N} y_{jt}+\sum_{k=1}^K \beta_k z_{k,it} + \sum_{\delta=1}^{\Delta} g_\delta(x_{\delta_{it}}) + f_1(s_{1i})+f_2(s_{2i})+f_{\tau}(\tau_t)+ \\ f_{1,2}(s_{1i},s_{2i})+f_{1,\tau}(s_{1i},\tau_t)+f_{2,\tau}+(s_{2i},\tau_t)+f_{1,2,\tau}(s_{1i},s_{2i},\tau_t)+\epsilon_{it}\]

\[\epsilon_{it} \sim i.i.d.(0,\sigma^2_\epsilon)\]

    sp3danovasarar1 <- pspatfit(form3d_psanova_restr, data = unemp_it,
                                  listw = lwsp_it, method = "Chebyshev",
                                  type = "sar", cor = "ar1")
    summary(sp3danovasarar1)

    ##
    ##  Call
    ## pspatfit(formula = form3d_psanova_restr, data = unemp_it, listw = lwsp_it,
    ##     type = "sar", method = "Chebyshev", cor = "ar1")
    ##
    ##  Parametric Terms
    ##                    Estimate Std. Error t value  Pr(>|t|)
    ## (Intercept)       4.027011   7.811451  0.5155   0.60623
    ## partrate          0.152078   0.021345  7.1247 1.389e-12 ***
    ## agri             -0.024655   0.022055 -1.1179   0.26372
    ## cons             -0.017631   0.042885 -0.4111   0.68102
    ## f1_main.1         0.196195  12.177654  0.0161   0.98715
    ## f2_main.1       -12.570282   9.946143 -1.2638   0.20642
    ## ft_main.1         1.193027   0.843386  1.4146   0.15733
    ## f12_int.1        -5.807022  11.115730 -0.5224   0.60143
    ## f12t_int.1        0.502141   2.523234  0.1990   0.84228
    ## pspl(serv)        0.180237   0.165103  1.0917   0.27509
    ## pspl(empgrowth)  -0.316471   0.053493 -5.9161 3.788e-09 ***
    ## rho               0.054391   0.021645  2.5128   0.01204 *
    ## phi               0.334044   0.013651 24.4705 < 2.2e-16 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ##  Non-Parametric Terms
    ##                     EDF
    ## pspl(serv)      4.0400
    ## pspl(empgrowth) 2.4343
    ##
    ##  Non-Parametric Spatio-Temporal Trend
    ##         EDF
    ## f1   10.197
    ## f2    9.761
    ## ft    7.879
    ## f12  42.100
    ## f1t   0.000
    ## f2t   0.000
    ## f12t 79.119
    ##
    ##  Goodness-of-Fit
    ##
    ##  EDF Total: 167.53
    ##  Sigma: 1.62877
    ##  AIC:  5495.29
    ##  BIC:  6469.11

Examples of LR test:

    anova(gam, gamsar, lrtest = TRUE)

    ##         logLik rlogLik    edf    AIC    BIC LRtest p.val
    ## gam    -4716.3 -4732.1 11.731 9456.1 9555.7
    ## gamsar -3867.7 -3885.9 13.917 7763.3 7880.4 1692.3     0

    anova(gam, gamsar, lrtest = TRUE)

    ##         logLik rlogLik    edf    AIC    BIC LRtest p.val
    ## gam    -4716.3 -4732.1 11.731 9456.1 9555.7
    ## gamsar -3867.7 -3885.9 13.917 7763.3 7880.4 1692.3     0

    anova(gamsar, gamsem, lrtest = FALSE)

    ##         logLik rlogLik    edf    AIC    BIC
    ## gamsar -3867.7 -3885.9 13.917 7763.3 7880.4
    ## gamsem -4024.6 -4041.2 16.836 8082.9 8213.7

    anova(gam, sp3danova_restr, lrtest = TRUE)

    ##                  logLik rlogLik     edf    AIC    BIC LRtest p.val
    ## gam             -4716.3 -4732.1  11.731 9456.1 9555.7
    ## sp3danova_restr -2802.8 -2807.8 206.837 6019.3 7213.4 3848.6     0

    anova(sp3danova_restr, sp3danovasarar1, lrtest = FALSE)

    ##                  logLik rlogLik    edf    AIC    BIC
    ## sp3danova_restr -2802.8 -2807.8 206.84 6019.3 7213.4
    ## sp3danovasarar1 -2580.1 -2587.3 167.53 5495.3 6471.6

Plot of non-parametric terms

    list_varnopar <- c("serv", "empgrowth")
    terms_nopar <- fit_terms(sp3danova_restr, list_varnopar)
    names(terms_nopar)

    ## [1] "fitted_terms"           "se_fitted_terms"
    ## [3] "fitted_terms_fixed"     "se_fitted_terms_fixed"
    ## [5] "fitted_terms_random"    "se_fitted_terms_random"

    plot_terms(terms_nopar, unemp_it, alpha = 0.05)

### 1.3.1 Parametric and nonparametric direct, indirect and total impacts

    imp_nparvar <- impactspar(sp3danovasarar1, listw = lwsp_it)
    summary(imp_nparvar)

    ##
    ##  Total Parametric Impacts (sar)
    ##           Estimate Std. Error   t value Pr(>|t|)
    ## partrate  0.161421   0.022624  7.135004   0.0000
    ## agri     -0.025590   0.023074 -1.109062   0.2674
    ## cons     -0.015641   0.045511 -0.343672   0.7311
    ##
    ##  Direct Parametric Impacts (sar)
    ##           Estimate Std. Error   t value Pr(>|t|)
    ## partrate  0.152727   0.021223  7.196361   0.0000
    ## agri     -0.024216   0.021834 -1.109105   0.2674
    ## cons     -0.014781   0.042970 -0.343994   0.7309
    ##
    ##  Indirect Parametric Impacts (sar)
    ##             Estimate  Std. Error     t value Pr(>|t|)
    ## partrate  0.00869470  0.00372253  2.33569690   0.0195
    ## agri     -0.00137355  0.00142789 -0.96194786   0.3361
    ## cons     -0.00085948  0.00274455 -0.31315901   0.7542

    imp_nparvar <- impactsnopar(sp3danovasarar1, listw = lwsp_it, viewplot = TRUE)

### 1.3.2 Spatial trend 3d (ANOVA).

Plot of spatial trends in 1996, 2005 and 2019

    plot_sp3d(sp3danovasarar1, data = unemp_it_sf,
              time_var = "year", time_index = c(1996, 2005, 2019),
              addmain = FALSE, addint = FALSE)

Plot of spatio-temporal trend, main effects and interaction effect for a year:

    plot_sp3d(sp3danovasarar1, data = unemp_it_sf,
              time_var = "year", time_index = c(2019),
              addmain = TRUE, addint = TRUE)

Plot of temporal trend for each province:

    plot_sptime(sp3danovasarar1, data = unemp_it,
                time_var = "year", reg_var = "prov")

Plots of fitted and residuals of the last model:

    fits <- fitted(sp3danovasarar1)
    resids <- residuals(sp3danovasarar1)
    plot(fits, unemp_it$unrate)

    plot(fits, resids)

# References

Mı́nguez, Román, Roberto Basile, and Marı́a Durbán. 2020. “An Alternative Semiparametric Model for Spatial Panel Data.” Statistical Methods & Applications 29 (4): 669–708. https://doi.org/10.1007/s10260-019-00492-8.