Getting lagged values of indep. variables to model contemporaneous values of the dep. variable

Question

I am trying to forecast the variable, oenb_dependent:

My current sample data looks like that:

> dput(datSel)
structure(list(oenb_dependent = c(1.0227039, -5.0683144, 0.6657713, 
3.3161374, -2.1586704, -0.7833623, -0.2203209, 2.416144, -1.7625406, 
-0.1565037, -7.9803936, 9.4594715, -4.8104584, 8.4827107, -6.1895262, 
1.4288595, 1.4896459, -0.4198522, -5.1583964, 5.2502294, 1.0567102, 
-1.0923342, -1.5852298, 0.6061936, -0.3752335, 2.5008664, -1.3999729, 
2.2802166, -2.1468756, -1.4890328, -0.79254376, 3.21804705, -0.94407886, 
-0.27802316, -0.20753079, -1.12610048, 2.0883735, -0.7424854, 
0.44203729, -1.48905938, 1.39644424, -3.8917377, 11.25665848, 
-9.22884035, 3.26856762, -0.00179541, -2.39664325, 4.00455574, 
-5.60891295, 4.6556348, -4.40536951, 6.64234497, -7.34787319, 
7.56303006, -8.23083674, 4.43247855, 1.31090412), carReg = c(0.73435946, 
0.24001161, 16.90532537, -14.60281976, 6.47603166, -8.35815849, 
3.55576685, 7.10705794, -4.6955223, 10.9623709, 5.5801857, -6.4499936, 
-9.46196502, 9.36289122, -8.52630424, 5.45070994, -4.5346405, 
-2.26716538, 2.56870398, 0.013737, 5.7750101, -27.1060826, 1.08977179, 
4.94934712, 17.55391859, -13.91160577, 10.38981128, -11.81349246, 
-0.0831467, 2.79748237, 1.84865463, -1.98736934, -6.24191695, 
13.33602659, -3.86527871, 0.78720993, 4.73360651, -4.1674034, 
9.37426802, -5.90660464, -0.4915792, -5.84811629, 9.67648643, 
-6.96872719, -7.6535767, 0.24847595, 0.18685263, -2.28766949, 
1.1544631, -3.87636933, -2.4731545, 4.33876671, 1.08836339, 5.64525271, 
1.90743854, -3.94709355, -0.84611324), cpi = c(1.16, -3.26, 0.22, 
-3.51, 0.84, -2.81, -0.34, -4.57, -0.12, -3.95, -1.37, -2.73, 
0.35, -5.38, -4.43, -3.08, 0.74, -3.03, -1.09, -2, 0.35, -1.52, 
1.28, 0.2, -0.25, -4.55, -2.49, -4.24, -0.31, -2.96, -2.24, -0.46, 
-0.06, -2.67, -1.27, -1.4, -0.7, -0.96, -2.18, -2.53, -0.52, 
-1.74, -2.18, -1.4, -0.34, -0.09, -1.65, -1.15, -0.17, -2.01, 
-1.38, -1.24, 0.09, -2.44, -1.92, -2.61, -0.34), primConstTot = c(-0.33334, 
-0.93333, -0.16667, -0.33333, -0.16667, -0.86666, -0.3, -0.4, 
-0.26667, -1.56667, -0.73333, 0.1, -0.23333, -0.26667, -1.5774, 
-0.19284, 0.38568, -2.42423, -0.93663, 0.08265, -0.63361, 0.0551, 
-0.49587, 2.39668, -1.70798, -3.36085, -2.56196, 0.16529, 0, 
-1.84572, -1.3774, -0.49586, -1.70798, -1.90081, -0.55096, -0.77134, 
-0.16529, -0.30303, -0.17066, -0.23853, -0.64401, -1.52657, -1.57426, 
-0.28623, -0.54861, -1.07336, -0.71558, 0.02385, -0.38164, -1.09721, 
0, 0.14311, -0.38164, -1.02566, -0.42934, -0.35779, -0.4532), 
    resProp.Dwell = c(0.8, -4, -3.2, 2.7, -1.6, -1, -2.4, -0.4, 
    -0.8, 1, -12.1, 0.2, -5.2, 3.7, -2.7, -1.7, 1.5, 0.7, -7.9, 
    0.3, 0.3, 1.4, -3.3, -1, -1.6, 1.5, 0.5, 1.5, -1, -2.2, -3.5, 
    0.5, 0.5, -0.9, -0.4, -3.4, 0.9, 0.1, -0.2, -2.8, -0.8, -6.2, 
    11.3, -4.6, 1, 1.1, -1.7, 4.1, -5, 2.3, -2.3, 4.6, -6.3, 
    6.3, -6.9, 0, 2.4), cbre.office.primeYield = c(0, 0, 0.15, 
    0.15, 0.2, 0.2, 0.2, 0.25, 0.25, 0.25, 0.25, 0.2, 0.15, 0.1, 
    0.05, 0.15, 0.3, 0.35, 0.4, 0.3, 0.2, 0, -0.15, -0.85, -1, 
    -0.85, -0.75, -0.1, 0, 0, 0, 0.05, 0.05, 0.05, 0.05, 0, 0, 
    0, 0.2, 0.2, 0.2, 0.2, 0, 0, 0, 0, 0.25, 0.25, 0.25, 0.25, 
    0, 0, 0, 0, 0, 0, 0), cbre.retail.capitalValue = c(-1882.35294, 
    230.76923, -230.76923, -226.41509, -670.78117, -436.13707, 
    -222.22223, 0, -205.91233, -202.16847, 0, -393.5065, -403.91909, 
    -186.30647, -539.81107, -748.11463, -764.70588, -311.47541, 
    -301.42782, -627.09677, -480, 720, 782.6087, 645.96273, 251.42857, 
    1386.66667, -533.33334, -533.33333, -533.33333, 0, 0, -1024.56141, 
    -192.10526, 0, -730, 0, 0, 0, 0, 0, -834.28571, 0, -1450.93168, 
    0, 0, 0, -700.78261, 0, 0, 0, 0, 0, 0, 0, -1452, 0, 0)), .Names = c("oenb_dependent", 
"carReg", "cpi", "primConstTot", "resProp.Dwell", "cbre.office.primeYield", 
"cbre.retail.capitalValue"), row.names = c(NA, -57L), class = "data.frame")

To forecast a dependent variable, works best, if I have lagged values of independent variables.(Many thx to the user Richard Hardy, who brought me on this idea!) So,

$$y_t=X_{t−k}β+ε_t$$

In this model, lagged values of $X_t$ are used to model contemporaneous values of $y_t$. Hence, given such a model, "predictions" of $y_t$ are available k periods ahead: $\hat y_{t+h}=X_{t-k+h} \beta$, and $X_{t-k+h}$ is available (as of time $t$) for all $k⩽h$.

Any suggestion, how to find such dependent variable. Which statistical methods are suitable for this problem?

I really would appreciate examples!

Answer 1

From the subject-matter perspective, you should look for leading indicators. Is there a variable/indicator (economic or financial, or whatever) that reacts first to shocks that only later affect the variable you intend to forecast? That is, is there a variable that moves first, and subsequently you see a corresponding move in the variable to be forecast? If there is, use it as an independent variable.

From the statistical perspective, you may look at cross-correlations between different variables and the variable you are going to forecast to see which ones are leading the one of interest. Finding high cross-correlations at lags other than zero is your goal (of course, you are interested in the cases where the other variables lead the one of interest, and not the other way around). You may use functions ccf (applied on two variables) and acf (applied on a vector-valued variable) in R.