Modern econometrics was articulated by studies of Trygve Haavelmo, who won Nobel Memorial Prize in Economic Sciences in 1989. He asserted that quantitative economic models must be probability models (or stochastic ones). Economic models should incorporate randomness, but deterministic models should not contain stochastic errors (these are systems in which no randomness is involved). Economic models contain some randomness that is why the best way to study them is by applying powerful theories of statistics.
Structural approach is the closest to Haavelmo’s original idea. A probabilistic economic model is specified, and analyses are performed under the assumption that economic model is correctly specified. This model is more viewed as an approximation. So this econometric analysis can be done under different interpretations. Quasistructural approach view this econometric model as an approximation rather than as a truth. This theory has led to concepts like :quasi-likelihood function, quasi-MLE. Semiparametric approach is an econometric model that is partially specified, but some features are left still unspecified. This type of studying implies using estimation methods, as: least-square and Generalized Method of Moments. This type of approach dominates contemporary econometrics.
Another branch of quantitative structural economics is calibration approach. This kind of approach interprets structural models as approximations and rather as false ones.