Author: Sebastian Krantz
Date: 2023-04-01

# Introduction to dfms

#### Sebastian Krantz

#### 2023-04-01

dfms provides a user friendly and computationally efficient approach to estimate linear Gaussian Dynamic Factor Models in R. The package is not geared at any specific application, and can be used for dimensionality reduction, forecasting and nowcasting systems of time series. The use of the package is facilitated by a comprehensive set of methods to explore/plot models and extract results.

    library(dfms)
    library(xts)

This vignette walks through the main features of the package. The data provided in the package in xts format is taken from Banbura and Modugno (2014)1, henceforth BM14, and covers the Euro Area from January 1980 through September 2009.

    # Using the monthly series from BM14
    dim(BM14_M)
    #> [1] 357  92
    range(index(BM14_M))
    #> [1] "1980-01-31" "2009-09-30"
    head(colnames(BM14_M))
    #> [1] "ip_total"       "ip_tot_cstr"    "ip_tot_cstr_en" "ip_constr"
    #> [5] "ip_im_goods"    "ip_capital"
    plot(scale(BM14_M), lwd = 1)

The data frame `BM14_Models` provides information about the series2, and the various models estimated by BM14.

    head(BM14_Models, 3)
    #>           series
    #> 1       ip_total
    #> 2    ip_tot_cstr
    #> 3 ip_tot_cstr_en
    #>                                                                                           label
    #> 1                                       IP-Total Industry - Working Day and Seasonally Adjusted
    #> 2              IP-Total Industry (Excluding Construction) - Working Day and Seasonally Adjusted
    #> 3 IP-Total Industry Excluding Construction and MIG Energy - Working Day and Seasonally Adjusted
    #>                           code freq log_trans small medium large
    #> 1 sts.m.i5.Y.PROD.NS0010.4.000    M      TRUE FALSE  FALSE  TRUE
    #> 2 sts.m.i5.Y.PROD.NS0020.4.000    M      TRUE  TRUE   TRUE  TRUE
    #> 3 sts.m.i5.Y.PROD.NS0021.4.000    M      TRUE FALSE  FALSE  TRUE

    # Using only monthly data
    BM14_Models_M <- subset(BM14_Models, freq == "M")

Prior to estimation, all data is differenced by BM14, and some series are log, differenced, as indicated by the `log_trans` column in `BM14_Models`. In general, dfms uses a stationary Kalman Filter with time-invariant system matrices, and therefore expects data to be stationary. Data is also scaled and centered3 in the main `DFM()` function, thus this does not need to be done by the user.

    library(magrittr)
    # log-transforming and first-differencing the data
    BM14_M[, BM14_Models_M$log_trans] %<>% log()
    BM14_M_diff = diff(BM14_M)
    plot(scale(BM14_M_diff), lwd = 1)

## Determining the Structure of the Model

Before estimating a model, the `ICr()` function can be applied to determine the number of factors. It computes 3 information criteria proposed in Bai and NG (2002)4, whereby the second criteria generally suggests the most parsimonious model.

    ic = ICr(BM14_M_diff)
    #> Missing values detected: imputing data with tsnarmimp() with default settings
    print(ic)
    #> Optimal Number of Factors (r) from Bai and Ng (2002) Criteria
    #>
    #> IC1 IC2 IC3
    #>   7   7  13
    plot(ic)

Another option is to use a Screeplot to gauge the number of factors by looking for a kink in the plot. A mathematical procedure for finding the kink was suggested by Onatski (2010)5, but this is currently not implemented in `ICr()`.

    screeplot(ic)

Based on both the information criteria and the Screeplot, I gauge that a model with 4 factors should be estimated, as factors, 5, 6 and 7 do not add much to the explanatory power of the model. Next to the number of factors, the lag order of the factor-VAR of the transition equation should be estimated (the default is 1 lag). This can be done using the `VARselect()` function from the vars package, with PCA factor estimates reported by `ICr()`.

    # Using vars::VARselect() with 4 principal components to estimate the VAR lag order
    vars::VARselect(ic$F_pca[, 1:4])
    #> $selection
    #> AIC(n)  HQ(n)  SC(n) FPE(n)
    #>      6      3      3      6
    #>
    #> $criteria
    #>                 1          2          3          4          5          6
    #> AIC(n)   5.810223   5.617282   5.427760   5.389413   5.407765   5.381829
    #> HQ(n)    5.898758   5.776646   5.657953   5.690434   5.779614   5.824507
    #> SC(n)    6.032560   6.017490   6.005838   6.145361   6.341582   6.493517
    #> FPE(n) 333.696100 275.153456 227.671078 219.144228 223.265640 217.639133
    #>                 7          8          9         10
    #> AIC(n)   5.409877   5.394900   5.421375   5.460761
    #> HQ(n)    5.923383   5.979235   6.076538   6.186753
    #> SC(n)    6.699434   6.862328   7.066673   7.283929
    #> FPE(n) 223.956824 220.793226 226.933863 236.331677

The selection thus suggests we should estimate a factor model with `r = 4` factors and `p = 3` lags6. Before estimating the model I note that dfms does not deal with seasonality in series, thus it is recommended to also seasonally adjust data, e.g. using the seasonal package before estimation. BM14 only use seasonally adjusted series, thus this is not necessary with the example data provided.

## Estimation and Exploration

Estimation can then simply be done using the `DFM()` function with parameters `r` and `p`7.

    # Estimating the model with 4 factors and 3 lags using BM14's EM algorithm
    model1 = DFM(BM14_M_diff, r = 4, p = 3)
    #> Converged after 26 iterations.
    print(model1)
    #> Dynamic Factor Model: n = 92, T = 356, r = 4, p = 3, %NA = 25.8366
    #>
    #> Factor Transition Matrix [A]
    #>      L1.f1   L1.f2   L1.f3   L1.f4   L2.f1   L2.f2   L2.f3   L2.f4   L3.f1
    #> f1  0.4720 -0.1297  0.8460  0.2098 -0.0733 -0.1436 -0.0595  0.1565  0.2356
    #> f2 -0.1612  0.1699  0.2389  0.1598  0.0641 -0.1341 -0.0542  0.1287  0.1336
    #> f3  0.3965  0.3264  0.0213 -0.3033 -0.1542 -0.0467 -0.1484 -0.0150 -0.1172
    #> f4  0.1096  0.1601 -0.1578  0.2485 -0.0365 -0.0563 -0.0230 -0.1117 -0.0719
    #>      L3.f2   L3.f3   L3.f4
    #> f1 -0.0803 -0.0386  0.0408
    #> f2  0.1347 -0.0024 -0.0342
    #> f3 -0.0087  0.1767  0.0249
    #> f4  0.0307  0.0662 -0.0035
    plot(model1)

The model can be investigated using `summary()`, which returns an object of class ‘dfm_summary’ containing the system matrices and summary statistics of the factors and the residuals in the measurement equation, as well as the R-Squared of the factor model for individual series. The print method automatically adjusts the amount of information printed to the data size. For large databases with more than 40 series, no series-level statistics are printed.

    dfm_summary <- summary(model1)
    print(dfm_summary) # Large model with > 40 series: defaults to compact = 2
    #> Dynamic Factor Model: n = 92, T = 356, r = 4, p = 3, %NA = 25.8366
    #>
    #> Call:  DFM(X = BM14_M_diff, r = 4, p = 3)
    #>
    #> Summary Statistics of Factors [F]
    #>       N     Mean   Median      SD       Min      Max
    #> f1  356  -0.0448   0.3455  4.4505  -21.9265  11.0306
    #> f2  356  -0.0319  -0.0892    2.68   -9.9549   7.4988
    #> f3  356  -0.1032  -0.0593  3.2891  -12.0969  16.2455
    #> f4  356  -0.0118    0.089   2.161   -8.2883  10.7219
    #>
    #> Factor Transition Matrix [A]
    #>      L1.f1   L1.f2    L1.f3   L1.f4    L2.f1    L2.f2    L2.f3    L2.f4   L3.f1
    #> f1  0.4720 -0.1297  0.84605  0.2098 -0.07334 -0.14356 -0.05950  0.15645  0.2356
    #> f2 -0.1612  0.1699  0.23889  0.1598  0.06406 -0.13413 -0.05415  0.12869  0.1336
    #> f3  0.3965  0.3264  0.02128 -0.3033 -0.15424 -0.04669 -0.14839 -0.01495 -0.1172
    #> f4  0.1096  0.1601 -0.15776  0.2485 -0.03655 -0.05626 -0.02304 -0.11169 -0.0719
    #>        L3.f2     L3.f3     L3.f4
    #> f1 -0.080320 -0.038592  0.040812
    #> f2  0.134692 -0.002391 -0.034215
    #> f3 -0.008694  0.176663  0.024876
    #> f4  0.030716  0.066201 -0.003465
    #>
    #> Factor Covariance Matrix [cov(F)]
    #>           f1        f2        f3        f4
    #> f1  19.8067    2.0846*  -3.4700*  -2.1094*
    #> f2   2.0846*   7.1822   -2.8725*  -1.0631*
    #> f3  -3.4700*  -2.8725*  10.8182    1.9286*
    #> f4  -2.1094*  -1.0631*   1.9286*   4.6701
    #>
    #> Factor Transition Error Covariance Matrix [Q]
    #>         u1      u2      u3      u4
    #> u1  9.0178  0.3303 -3.0764 -1.0182
    #> u2  0.3303  5.4425 -1.3095 -0.5051
    #> u3 -3.0764 -1.3095  7.0230  0.8639
    #> u4 -1.0182 -0.5051  0.8639  3.8005
    #>
    #> Summary of Residual AR(1) Serial Correlations
    #>    N     Mean   Median      SD      Min     Max
    #>   92  -0.0409  -0.0782  0.2959  -0.5073  0.6858
    #>
    #> Summary of Individual R-Squared's
    #>    N    Mean  Median      SD     Min     Max
    #>   92  0.3712   0.299  0.2888  0.0067  0.9978

    # Can request more detailed printouts
    # print(dfm_summary, compact = 1)
    # print(dfm_summary, compact = 0)

Apart from the model summary, the dfm methods `residuals()` and `fitted()` return observation residuals and fitted values from the model. The default format is a plain matrix, but the functions also have an argument to return data in the original (input) format.

    plot(resid(model1, orig.format = TRUE))

    plot(fitted(model1, orig.format = TRUE))

Another way to examine the factor model visually is to plot the Quasi-Maximum-Likelihood (QML) factor estimates against PCA and Two-Step estimates following Doz, Giannone and Reichlin (2011)8, where the Kalman Filter and Smoother is run only once. Both estimates are also computed by `DFM()` during EM estimation and can also be visualized with `plot.dfm`.

    plot(model1, method = "all", type = "individual")

The plot with the various estimates shows that the QML estimates are more volatile in the initial periods where there are many missing series, but less volatile in the latter periods. In general, QML estimates may now always be superior across the entire data range to Two-Step and PCA estimates. Often, Two-Step estimates also provide similar forecasting performance, and are much faster to estimate using `DFM(BM14_M_diff, r = 4, p = 3, em.method = "none")`.

The factor estimates themselves can be extracted in a data frame using `as.data.frame()`, which also provides various options regarding the estimates retained and the format of the frame. It is also possible to add a time variable from the original data (the default is a sequence of integers).

    # Default: all estimates in long format
    head(as.data.frame(model1, time = index(BM14_M_diff)))
    #>   Method Factor       Time      Value
    #> 1    PCA     f1 1980-02-29  1.1503013
    #> 2    PCA     f1 1980-03-31  0.3374613
    #> 3    PCA     f1 1980-04-30 -1.4905337
    #> 4    PCA     f1 1980-05-31 -1.5360036
    #> 5    PCA     f1 1980-06-30 -0.2828433
    #> 6    PCA     f1 1980-07-31  0.3385403

## Forecasting

DFM forecasts can be obtained with the `predict()` method, which dynamically forecasts the factors using the transition equation (default 10 periods), and then also predicts data forecasts using the observation equation. Objects are of class ‘dfm_forecast’

    # 12-period ahead DFM forecast
    fc = predict(model1, h = 12)
    print(fc)
    #> 12 Step Ahead Forecast from Dynamic Factor Model
    #>
    #> Factor Forecasts
    #>        f1     f2      f3      f4
    #> 1  3.4698 1.7276  0.2706 -0.9064
    #> 2  2.4379 0.0305  1.3119 -0.0097
    #> 3  1.6576 0.3182  0.7247  0.1037
    #> 4  1.7445 0.7829 -0.2230 -0.1513
    #> 5  0.8281 0.1390  0.5545  0.1199
    #> 6  0.9576 0.3001  0.0126 -0.0923
    #> 7  0.6599 0.2489  0.0333 -0.0518
    #> 8  0.3354 0.0681  0.1999  0.0332
    #> 9  0.4286 0.1824 -0.0830 -0.0672
    #> 10 0.1853 0.0605  0.0882  0.0107
    #> 11 0.1605 0.0544  0.0268 -0.0067
    #> 12 0.1509 0.0734 -0.0288 -0.0235
    #>
    #> Series Forecasts
    #>    ip_total ip_tot_cstr ip_tot_cstr_en ip_constr ip_im_goods ip_capital
    #>    ip_d_cstr ip_nd_cons   ip_en ip_en_2 ip_manuf ip_metals ip_chemicals
    #>    ip_electric ip_machinery ip_paper ip_plastic new_cars orders
    #>    ret_turnover_defl ecs_ec_sent_ind ecs_ind_conf ecs_ind_order_book
    #>    ecs_ind_stocks ecs_ind_prod_exp ecs_ind_prod_rec_m ecs_ind_x_orders
    #>    ecs_ind_empl_exp ecs_cons_conf ecs_cons_sit_over_next_12 ecs_cons_exp_unempl
    #>    ecs_cons_gen_last_12m ecs_cstr_conf ecs_cstr_order_books ecs_cstr_empl_exp
    #>    ecs_cstr_prod_recent ecs_ret_tr_conf ecs_ret_tr_bus_sit ecs_ret_tr_stocks
    #>    ecs_ret_tr_exp_bus ecs_ret_tr_empl ecs_serv_conf ecs_serv_empl_exp
    #>    pms_comp_output pms_comp_empl pms_pmi pms_manuf_empl pms_manuf_output
    #>    pms_manuf_product pms_serv_out pms_serv_empl pms_serv_new_bus
    #>    pms_serv_product     urx empl_total empl_tot_xc empl_cstr empl_manuf
    #>    extra_ea_trade_exp_val intra_ea_trade_exp_val extra_ea_trade_imp_val
    #>    intra_ea_trade_imp_val  us_ip  us_urx us_empl us_retail_sales
    #>    us_ip_manuf_exp us_cons_exp us_r3_m us_r10_year      m3   loans ir_long
    #>    ir_short ir_1_year ir_2_year ir_5_year    eer eer_cpi eer_ppi exr_usd
    #>    exr_gbp rxr_yen euro50 euro325  sp500  dow_j raw_mat_en raw_mat_oil
    #>    raw_mat_gold raw_mat_oil_fwd raw_mat
    #>  [ reached getOption("max.print") -- omitted 12 rows ]

These forecasts can also be visualized using a plot method. By default the entire series history is plotted along with the forecasts, thus it is often helpful to restrict the plot range. As with any stationary autoregressive model, the forecasts tend to zero quite quickly9.

    # Setting an appropriate plot range to see the forecast
    plot(fc, xlim = c(320, 370))

    # Predicting with Two-Step estimates
    # plot(predict(model1, h = 12, method = "2s"), xlim = c(320, 370))

The forecasts can also be retrieved in data frame using `as.data.frame()`. Again the method has various arguments to control the forecasts retained (factors, data or both, default factors), and the format of the frame.

    # Factor forecasts in wide format
    head(as.data.frame(fc, pivot = "wide"))
    #>   Time Forecast         f1         f2         f3          f4
    #> 1    1    FALSE   4.207651 -1.6552707  3.2190834 -1.82663141
    #> 2    2    FALSE  -3.656458 -6.6178403 -3.9121446 -0.05762001
    #> 3    3    FALSE -14.486337 -0.6573393 -6.8663850  1.69525122
    #> 4    4    FALSE -14.794458  3.1573790 -6.4820328  1.70916785
    #> 5    5    FALSE  -8.191696  0.5563773 -0.1840816  0.98407541
    #> 6    6    FALSE  -1.357927 -0.3805697  3.5068964 -0.05346754

## Estimation with Mixed Frequency

dfms currently provides no specific adjustments for data at different frequencies. An algorithm that accommodates monthly and quarterly series is planned for summer 2023. In the meantime, users may choose to block the data (creating multiple quarterly series from a monthly series, and duplicating quarterly series to maintain equal representation).

## Additional Functions

dfms also exports central functions that help with DFM estimation, such as imputing missing values with `tsnarmimp()`, estimating a VAR with `.VAR()`, or Kalman Filtering and Smoothing with `SKFS()`, or separately with `SKF()` followed by `FIS()`. To my knowledge these are the fastest routines for simple stationary Kalman Filtering and Smoothing currently available in R. The function `em_converged()` can be used to check convergence of the log-likelihood in EM estimation.

dfms also exports a matrix inverse and pseudo-inverse from the Armadillo C++ library through the functions `ainv()` and `apinv()`. These are often faster than `solve()`, and somewhat more robust in near-singularity cases.

## References

Doz, C., Giannone, D., & Reichlin, L. (2011). A two-step estimator for large approximate dynamic factor models based on Kalman filtering. Journal of Econometrics, 164(1), 188-205.

Doz, C., Giannone, D., & Reichlin, L. (2012). A quasi-maximum likelihood approach for large, approximate dynamic factor models. Review of Economics and Statistics, 94(4), 1014-1024.

Banbura, M., & Modugno, M. (2014). Maximum likelihood estimation of factor models on datasets with arbitrary pattern of missing data. Journal of Applied Econometrics, 29(1), 133-160.

Mariano, R. S., & Murasawa, Y. (2003). A new coincident index of business cycles based on monthly and quarterly series. Journal of Applied Econometrics, 18(4), 427-443.

Bai, J., Ng, S. (2002). Determining the Number of Factors in Approximate Factor Models. Econometrica, 70(1), 191-221.

Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 92(4), 1004-1016.

Stock, J. H., & Watson, M. W. (2016). Dynamic Factor Models, Factor-Augmented Vector Autoregressions, and Structural Vector Autoregressions in Macroeconomics. Handbook of Macroeconomics, 2, 415–525.

1. Banbura, M., & Modugno, M. (2014). Maximum likelihood estimation of factor models on datasets with arbitrary pattern of missing data. Journal of Applied Econometrics, 29(1), 133-160.↩︎

2. Both about the monthly ones and quarterly series contained in `BM14_Q`. The order of rows in `BM14_Models` matches the column order of series in `merge(BM14_M, BM14_Q)`.↩︎

3. Data must be scaled and centered because the Kalman Filter has no intercept term.↩︎

4. Bai, J., Ng, S. (2002). Determining the Number of Factors in Approximate Factor Models. Econometrica, 70(1), 191-221. doi:10.1111/1468-0262.00273↩︎

5. Onatski, A. (2010). Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 92(4), 1004-1016.↩︎

6. Some authors like Doz, Giannone, and Reichlin (2012) additionally allow the number of transition error processes, termed ‘dynamic factors’ and given an extra parameter `q`, to be less than the number of ‘static factors’ `r`. I find this terminology confusing and the feature not very useful for most practical purposes, thus I have not implemented it in dfms.↩︎

7. Users can also choose from two different implementations of the EM algorithm using the argument `em.method`. The default is `em.method = "auto"`, which chooses the modified EM algorithm proposed by BM14 for missing data if `anyNA(X)`, and the plain EM implementation of Dos, Giannone and Reichlin (2012), henceforth DGR12, otherwise. The baseline implementation of DGR12 is typically more than 2x faster than BM14’s method, and can also be used with data that has a few random missing values (< 5%), but gives wrong results with many and systematically missing data, such as ragged edges at the beginning of end of the sample or series at different frequencies. With complete datasets, both `em.method = "DGR"` and `em.method = "BM"` gives identical factor estimates, and in this case `em.method = "DGR"` should be used.↩︎

8. Doz, C., Giannone, D., & Reichlin, L. (2011). A two-step estimator for large approximate dynamic factor models based on Kalman filtering. Journal of Econometrics, 164(1), 188-205.↩︎

9. Depending also on the lag-order of the factor-VAR. Higher lag-orders product more interesting forecast dynamics.↩︎