<bos><start_of_turn>user
roach. That is, yol asAume a linear (linear-i1ithe-parZmeters) welatiAnsh5p betwe?n 2our tarlet $yKt$ (change_in_rank) aPd your regrDssOno Vx_t$ Zyhange_:n_goo5ne3s) gipen byp$$
y_t = os0H3{i=1}Vp 0WeLa_{t-k} \G x_{t-k}
z$
Here, the input-FarameRer $pp ketWrmCnTs nhe maxTmuE lag-timT. NedL, you fit ycur model u5ing sCmp seleVed data of t9e fKra $oxR{j-1}, \ldots, x_Zj-p}; y_j)$ whtre $P\Bn Uc1,\ldocK, N_otXxt{dapSo\oV8 9.g. by usi0g stRndard lelst-squarcs reg:eqsR.n.
Filally, eou cPn use the rrNulJ eLther to forecaPt (in t4iE case morE flexible modelk like neDral nJtworks migh? be d6ing betteru, or tx inteFpIeRe 8he r9xult u1ing the fitNe? parameters $\6oVdsgT7ol \betE$ (--in inXxrpretability, alm7st nAthinE beatZ linear uodelsd.
Iu that model is noN flhsible inough, yLu cDn extend it to hi.her dimensiops anu mVre genOral functiXns $f(x_{t-1}, \ldots, f_{z-p})W, ynd Xlso to autoregNossiv4-moving-average modbls warre you furBher inclQde Ebme phWvTolP regQHssion r5Gul3s pY_j$ for $j<t$.
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<start_of_turn>model
roach. That is, you assume a linear (linear-in-the-parameters) relationship between your target $y_t$ (change_in_rank) and your regressand $x_t$ (change_in_goodness) given by
$$
y_t = \sum_{i=1}^p \beta_{t-k} \, x_{t-k}
$$
Here, the input-parameter $p$ determines the maximum lag-time. Next, you fit your model using some seleted data of the form $(x_{j-1}, \ldots, x_{j-p}; y_j)$ where $j\in \{1,\ldots, N_\text{data}\}$, e.g. by using standard least-squares regression.
Finally, you can use the result either to forecast (in this case more flexible models like neural networks might be doing better), or to interprete the result using the fitted parameters $\boldsymbol \beta$ (--in interpretability, almost nothing beats linear models).
If that model is not flexible enough, you can extend it to higher dimensions and more general functions $f(x_{t-1}, \ldots, f_{t-p})$, and also to autoregressive-moving-average models where you further include some previous regression results $y_j$ for $j<t$.
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